ORTHOGONAL SIGNAL ALIGNMENT WITH VARIABLE RATEAND POWER IN MIMO Y CHANNELS

by

Xueying Yuan

Submitted in partial fulfillment of the requirementsfor the degree of Master of Applied Science

at

Dalhousie UniversityHalifax, Nova Scotia

June 2016

c⃝ Copyright by Xueying Yuan, 2016

To My Parents and Family

ii

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Abbreviations and Symbols Used . . . . . . . . . . . . . . . . . . x

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Wireless Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Signal Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 MIMO Y Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 MAC Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 BC Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Bit Loading Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Water-filling Algorithm . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Discrete Bit Loading Algorithm . . . . . . . . . . . . . . . . . 12

1.5 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Chapter 2 Orthogonal Signal Space Alignment for MIMO Y Chan-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Related Literature Evaluation . . . . . . . . . . . . . . . . . . . . . . 17

2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Orthogonality Application . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 ML Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Orthogonality Constraints . . . . . . . . . . . . . . . . . . . . 222.3.3 Calculations of Precoding Vectors . . . . . . . . . . . . . . . . 232.3.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Modifications in Orthogonal Signaling . . . . . . . . . . . . . . . . . . 282.4.1 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.2 Time Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . 29

iii

2.4.3 Iterative Search for Optimized Orientation . . . . . . . . . . . 36

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 3 Bit Loading and Power Allocation . . . . . . . . . . . . 42

3.1 Rate and Power Adaptations . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Bit Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Constellation Mapping . . . . . . . . . . . . . . . . . . . . . . 463.3.2 Modulation Thresholds . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Sub-optimal Power Allocation . . . . . . . . . . . . . . . . . . . . . . 513.4.1 Sub-optimal Power Allocation . . . . . . . . . . . . . . . . . . 52

3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Appendix A: Average Symbol Energy in the Superposed Constellations 65

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

iv

List of Tables

3.1 The bit mapping for the combination of two 4-PAM symbols. . 49

3.2 The modulations and the corresponding thresholds for achievingthe aimed BER 10−3. . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 The relative power required for achieving the aimed BER 10−3

using each modulation scheme compared to the current transmitpower for each user pair. . . . . . . . . . . . . . . . . . . . . . 53

3.4 The relative power required for achieving the aimed BER 10−3

using next modulation level compared to the current modulationscheme for each user pair. . . . . . . . . . . . . . . . . . . . . . 54

v

List of Figures

1.1 (a) Traditional non-NC vs. (b) Traditional NC vs. (c) PNC. . 4

1.2 3-paired TWRC with SA. . . . . . . . . . . . . . . . . . . . . 5

1.3 The original 3-user MIMO Y channel model. . . . . . . . . . . 6

1.4 The three aligned dimensions for mutual symbols using SA ina 3-user MIMO Y channel in the MAC phase. . . . . . . . . . 7

1.5 Power allocation in the water-filling algorithm. . . . . . . . . . 11

1.6 The Hyper-cube signaling. . . . . . . . . . . . . . . . . . . . . 15

2.1 The proposed 3-user MIMO Y channel model. . . . . . . . . . 19

2.2 Possible coordinates along D12, D13 and D23 for received sym-bols at the relay in 3D. . . . . . . . . . . . . . . . . . . . . . . 20

2.3 The possible signal received at the relay through the signal sum-mation over the air. . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 The comparison between the BER performances of the originalMIMO Y channel scheme and the proposed scheme for differentuser pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 The throughput in the proposed scheme for different user pairs. 27

2.6 The BER performance of the proposed MIMOY channel schemewith power allocation. . . . . . . . . . . . . . . . . . . . . . . 29

2.7 The BER performance of the proposed MIMOY channel schemewith time scheduling. . . . . . . . . . . . . . . . . . . . . . . . 31

2.8 The relationship between the thresholds and the correspondingTS loss ratio in time scheduling. . . . . . . . . . . . . . . . . . 32

2.9 The throughput of the proposed MIMO Y channel scheme withtime scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.10 The BER performance of the proposed MIMOY channel schemewith time scheduling applied first and then followed by powerallocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.11 The BER performance of the proposed MIMOY channel schemewith power allocation applied first and then followed by timescheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

vi

2.12 The comparison of the throughput in the scheme using powerallocation and in the scheme using both power allocation & timescheduling with different thresholds. . . . . . . . . . . . . . . . 35

2.13 The ratio of reduced used TSs when applying time scheduling& power allocation compared to the one when applying powerallocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.14 The BER performance of the proposed scheme with six iterations. 38

2.15 The BER performance of the proposed scheme with differentapplied methods. . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.16 The BER performance of the proposed scheme with six itera-tions combined with power allocation and time scheduling. . . 39

3.1 The decoding and remapping for BPSK symbols. . . . . . . . 47

3.2 The decoding and remapping for 4-QAM symbols. . . . . . . . 48

3.3 The ambiguity occurring in decoding and remapping for onedimension of 16-QAM symbols. . . . . . . . . . . . . . . . . . 48

3.4 The decoding and remapping of 4-PAM symbols correspondingto 1-D in 16-QAM signalling. . . . . . . . . . . . . . . . . . . 49

3.5 The BER performance of the received symbols at the relay withdifferent modulations in AWGN channels. . . . . . . . . . . . 50

3.6 The BER comparison between the system with (a) the bit load-ing and the integrated bit loading and (b) power allocation. . 55

3.7 The throughput comparison between the system with the bitloading and the integrated bit loading and power allocation. . 56

A.1 The minimum distance between symbols in BPSK modulationused at users and in the corresponding ternary symbols at therelay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.2 The minimum distance between symbols in 4-QAM originalconstellation at users and in the corresponding superposed con-stellation at the relay. . . . . . . . . . . . . . . . . . . . . . . . 66

A.3 The minimum distance between symbols in 16-QAM originalconstellation at users and in the superposed constellation atthe relay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

vii

A.4 The minimum distance between symbols in 64-QAM originalconstellation at users and in the superposed constellation atthe relay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.5 The probability of superposed symbols at the relay when theequiprobable 4-QAM is deployed at the users. . . . . . . . . . 68

viii

Abstract

The multiple-input multiple-output (MIMO) Y channel employs signal align-

ment (SA) to improve spectral efficiency of wireless networks. In this system of

three nodes exchanging data via the relay, cooperation of terminals allows the data

transfer of six messages in two phases, medium access control (MAC) and broadcast,

each taking one time slot thus achieving up to three times the rate of the conven-

tional relaying system without SA. The antenna configuration in the original MIMO

Y channel imposes limitations on controlling the minimum distance between signal-

ing points representing network coded messages and this affects symbol error rate

performance. To mitigate this problem, this thesis proposes to add one redundant

antenna at every user and designs corresponding precoding vectors to obtain flexi-

bility in the selection of orthogonal signaling dimensions and to simplify decoding at

the relay. Additionally, the thesis develops adaptive power allocation and bit loading

to optimize the system performance.

ix

List of Abbreviations and Symbols Used

2D 2-dimensional

3D 3-dimensional

AF amplify-and-forward

ANC analog network coding

AWGN additive white Gaussian noise

BC broadcast

BER bit error rate

BPSK binary phase shift keying

CSI channel state information

DF decode-and-forward

EM electromagnetic

IA interference alignment

MA margin-adaptive

MAC medium access control

MAI multiple access interference

MATLAB R⃝ mathematical laboratory

MIMO multiple-input multiple-output

ML maximum likelihood

MMSE minimum mean square error

NC network coding

OFDM orthogonal frequency division multiplexing

PAM pulse amplitude modulation

PNC physical network coding

PSK phase shift keying

QAM quadrature amplitude modulation

QPSK quadrature phase shift keying

RA rate-adaptive

SA signal alignment

x

SER symbol error rate

SNR signal-to-noise ratio

TDMA time division multiple access

TS(s) time slot(s)

TWRC two-way relay channel

XOR exclusive or

ZF zero-forcing

xi

Acknowledgements

First, I would like to give my thanks to my supervisor Dr.Ilow, who has taught

me many things. I also would like to thank all colleagues in my research group:

Aasem Alyahya, Rashed Alsakarnah, Fadhel Alhumaidi, Md Sabbir Hussain, Mahdi

Attaran, Zichao Zhou and Hui Xiong for their generous help. Finally, I’d like to thank

my family and friends for their support and understanding.

xii

Chapter 1

Introduction

Recent developments in wireless communication system designs are driven by the

goal of achieving high utilization of radio resources like spectrum and power. With

the ever growing demands for more sophisticated wireless services, the ultimate ob-

jective is to support an increased number of users and higher data rates. To achieve

this goal, several new approaches have been proposed recently with the cooperation

of terminals, signal alignment (SA) and more advanced relaying strategies offering

some of promising breakthroughs. In this thesis, we focus on using SA in cooper-

ative relay networks with multiple-input multiple-output (MIMO) terminals, where

multiple transmissions take place simultaneously over a common broadcast communi-

cation medium. Specifically, in a multi-way communication network with three nodes

exchanging messages with each another through a relay, called the Y channel, this

thesis designs a signal processing scheme to obtain the flexibility in the selection of

orthogonal signaling dimensions and to simplify the decoding at the relay.

Because of the broadcast nature of the wireless medium, when multiple signals

are transmitted at the same time and frequency to increase bandwidth utilization,

multiple access interference (MAI) is inevitably introduced. The recently proposed

interference alignment (IA) has mitigated the problems of MAI presented in the

conventional medium access schemes [1] by aligning signals in time, frequency or

space. Specifically, IA confines MAI to a small signal dimension subspace and leaves

the majority of signaling dimensions for desired signals, which allows for efficient

utilization of bandwidth. On the other hand, SA utilizes the concept of signal space

management from IA, and intends to align two desired signals from two users as a

whole to perform detection and encoding at the relay.

In the MIMO Y channel system, deploying SA and network coding, the data

exchange of six messages among three users is completed in two time slots (TSs),

which significantly enhances the throughput of the network over time division multiple

1

2

access (TDMA). In the first TS, called the multiple access control (MAC) phase,

three users send all messages simultaneously to the relay, and in the second TS,

called the broadcast (BC) phase, the relay transmits messages to users also on a

single MIMO channel access [2]. In a MIMO Y channel, all dimensions in a signal

space at the relay are exploited for transmitting network coded signal representations

of three aligned messages with the latter corresponding to network coded symbols

exchanged between a pair of users. In the MIMO Y channel scheme, the signal

alignment is accomplished via precoding vectors at the user sides in the MAC phase

and the relay in the BC phase. The relay in the MAC and the BC phases calls for

working with three receive/transmit antennas, while the user terminals can operate

with the minimum of two antennas. The antenna configuration where the relay is

equipped with three antennas and each user is equipped with two antennas is called

the original MIMO Y channel. This system imposes limitations on controlling the

minimum distance between signaling points representing network coded messages,

and this affects symbol error rate performance.

To mitigate this problem, this thesis adds one redundant antenna at every user

and designs corresponding precoding vectors to obtain the flexibility in the selection

of orthogonal signaling dimensions and to simplify decoding at the relay. The thesis

proposes different designs and optimization of precoding vectors to improve the bit

error rate (BER) performance over the original scheme. Specifically, we apply power

allocation to distribute energy among the users based on their channel conditions.

In addition, iterative optimization of signaling dimensions and time scheduling of

transmissions according to channel states are employed to further improve the BER

performance. We also propose the deployment of quadrature amplitude modulation

(QAM) in the MIMO Y channel and present adaptive deployment of QAM through

bit loading techniques.

The remainder of this chapter is organized as follows: in sections 1.1, 1.2 1.3 and

1.4, we present the building blocks related to our research in more detail, namely, (i)

network coding, (ii) SA, (iii) the original MIMO Y channel, and (iv) the bit loading

techniques. Section 1.5 describes the thesis objectives, while Section 1.6 presents the

organization of the whole thesis.

3

1.1 Wireless Network Coding

In wireless networks, there are two kinds of relays: amplify-and-forward (AF) and

decode-and-forward (DF) [3]. Relays in AF mode just amplify the received signal

and retransmit it, while relays in DF mode decode the received signal to reduce noise

effects, and then transmit the re-encoded signal. In cooperative communications

considered in this thesis, DF relays are required in order to process the received data.

Network coding (NC) is a technique where signals sent from terminals are mixed

together at relays and the relays transmit the combined signals to terminals, which

improves the network throughput. Although NC was initially proposed in the field

of wire-line networks [4], there has been growing interest in the application of NC in

wireless communications, because the broadcast nature of the wireless medium brings

greater efficiency in increasing the system capacity than in wire-line systems.

On the other hand, if the relay can broadcast mixed messages, it can also receive

mixed messages to further improve the throughput. It is practical because the analog

signals sent from terminals are electromagnetic (EM) waves that can be naturally

superimposed in the air as one signal. The wireless NC using this scheme is divided

into two parts: physical-layer network coding (PNC) [5] [6] and analog-network coding

(ANC) [7] [8]. In both cases, the relay receives the mixed signal, but the difference is

that in PNC the relay uses DF strategy, while in ANC the relay employs AF strategy.

In this thesis, we choose to use PNC.

To show the throughput advantage of NC in wireless networks, we use the model

of a two-way relay channel (TWRC) [9] visualized in Fig. 1.1 [10]. There are two

terminals A and B intending to exchange packets a and b, and the link between them

is through a relay station R. In the conventional non-NC scheme as in Fig. 1.1(a),

terminal A and terminal B each needs one time-slot (TS) to send a packet to the

relay, and the relay requires two TSs to broadcast two packets. Hence, the exchange

of two packets between two terminals in this case requires four TSs. Fig. 1.1(b) shows

that this exchange of data can be accomplished in three TSs: after terminals send

their packets independently to the relay in two TSs, the relay employs NC technique

to combine (XOR in bit level) the received two packets and broadcasts it in the third

TS. The terminals receive the same combined packet but each of them can decode

the packet with the knowledge of its own transmitted one. When employing PNC or

4

ANC, the process only requires two TSs as illustrated in Fig. 1.1(c). Terminals A

and B simultaneously transmit their packets to the relay with in-the-air addition of

packets, and the relay re-broadcasts the mixed (or XORed representation of) packet

it receives.

(a) (b) (c)

Figure 1.1: (a) Traditional non-NC vs. (b) Traditional NC vs. (c) PNC.

1.2 Signal Alignment

Although the broadcast nature of the wireless channel boosts the capability of

NC to improve network throughput, it also brings interference when multiple users

transmit signals in the same frequency band at the same time without orthogonal

multiplexing. Many investigations have been made and many approaches have been

developed to handle this interference. Recently, the IA proposed by Jafar et al. [11]–

[13] has drawn increasing attention to combat the MAI problem. IA is a cooperative

interference management strategy that attempts to align interfering signals in time,

frequency, or space. Its basic concept is to overlap interference signals to a small

signaling subspace at the receiver via linear precoding vectors at users and to let the

desired signals utilize most of the channel resources.

IA provides insight into the signal dimension spatial management and has led

to the research of feasibility of SA [14] [15]. In SA, every signal is meaningful for

some terminals so that all signal dimension subspace is available for desired signals,

which results in higher efficiency than IA. SA focuses on aligning signals with mutual

interests to the same dimension for combining them as a whole to proceed with

detection and encoding at the relay. For example, Fig. 1.2 shows a six-user TWRC

5

Figure 1.2: 3-paired TWRC with SA.

model where the terminals, indexed by j and j′ (j, j′ ∈ {1, 2, 3}) and each equipped

with one antenna, are paired into three groups denoted by different colors. Between

each pair, two packets intend to be exchanged via the relay station equipped with

three antennas. Precoding vectors, also called beamforming vectors, are carefully

selected by paired users so that the precoding vectors combined with the channel

matrix can align the signals from paired users into one spatial dimension. Hence,

the six messages sent by the six terminals are aligned in three spatial dimensions

denoted by colors and combined as three aligned signals which are received at the

relay. Then the relay decodes and re-encodes the received signals, so that different

colored signals are in different dimensions after being broadcast by the relay. Finally,

each terminal can receive signals on the selected dimensions that carry the desired

signal and decode the desired information. As this example shows, SA provides

superior bandwidth efficiency by exchanging six messages within two TSs. Moreover,

because the six signals are aligned into three signals in the air, three antennas are

required at the relay, compared with using the MIMO system without NC, where six

antennas are needed for receiving six signals.

1.3 MIMO Y Channel Model

The MIMO Y channel proposed by Lee [14] is a generalized model of a wireless

TWRC with three users. In this system, the users and the relay are equipped with

Nu and Nr antennas respectively, and the antenna configuration must satisfy the

condition 2Nu−Nr ≥ 1 to meet the SA requirements. Fig. 1.3 illustrates the minimum

antenna configuration MIMO Y channel with three user nodes indexed by i (i ∈

6

{1, 2, 3}) equipped with two antennas and one relay station equipped with three

antennas. Each user i has two messages for the other users, denoted as bj,i (j ∈{1, 2, 3}, j ̸= i) and bj,i ∈ {1, 0} at the bit level. If BPSK modulation is applied, the

symbol representation of bj,i is sj,i ∈ {1,−1}. The traditional assumptions for MIMO

Y channel is that all nodes work in half duplex mode and the perfect global channel

state information (CSI) is available at all nodes. When users transmit messages to

the relay, it is called the MAC phase; and when the relay broadcasts signals to users,

it is called the BC phase. The six messages are exchanged within two TSs in MIMO

Y channels.

Figure 1.3: The original 3-user MIMO Y channel model.

1.3.1 MAC Phase

In the MAC phase, each user selects two precoding vectors for processing their two

transmit messages, and then all users convey a total of six processed messages to the

relay simultaneously. Precoding vectors and the channel matrices will align the two

mutual signals to a dimension in the signal space, so that the received signal at the

relay is actually a superposed signal of three aligned signals in different dimensions.

Then the relay decodes three aligned signals with DF strategies.

Fig. 1.4 demonstrates the obtained aligned dimensions through SA which are

denoted byD12, D13 andD23, and the subscripts represent that this aligned dimension

is for mutual symbols between the corresponding users. For instance, D12 is the

7

aligned dimension for mutual symbols s2,1 and s1,2. The aligned dimensions are

determined by the product of the channel gain matrix Hr,i and the precoding vector

vj,i. Hr,i is a 3 × 2 matrix with each entry of Hr,i(t, f) indicating the gain from

transmit antenna f of the i-th terminal to the receive antenna t of the relay R. The

colors in Fig. 1.4 represent different users.

Figure 1.4: The three aligned dimensions for mutual symbols using SA in a 3-userMIMO Y channel in the MAC phase.

Two information symbols sl,i and sp,i (l, p = {1, 2, 3}; l, p ̸= i) are pre-processed

by the precoding matrix vi = [vl,i | vp,i] of size 2 × 2 at user i where the precoding

vectors vl,i and vp,i are of size 2 × 1. Then the transmitted data by terminal i is

vi ·[ sl,i

sp,i

]and the received signal from user i at the relay is Hr,i · vi ·

[ sl,ksp,k

].

The common power assumptions for the MIMO Y channel model are that (i) the

average power of information symbols is one and (ii) the total transmit power in the

system is constrained to one. The average transmit power Pi for one user can be

calculated as:

Pi= E{(vl,i ·sl,i)H· (vl,i ·sl,i) + (vp,i ·sp,i)H·vp,i ·sp,i)

}= E

{vHl,i · vl,i + vHp,i · vp,i

} (1.1)

where E{x} is the expectation of random variable x. As indicated in (1.1), when the

average power of information symbols is constrained to one, the transmit power is

only defined by the square of norms of precoding vectors. In addition, we assume the

sum power for transmitting a pair of mutual symbols are the same, so the transmit

8

power for the whole system PT can be calculated as:

PT =3∑

i=1

{vHl,i · vl,i + vHp,i · vp,i

}= |v2,1|2 + |v3,1|2 + |v1,2|2 + |v3,2|2 + |v1,3|2 + |v2,3|2

= (|v2,1|2 + |v1,2|2) + (|v3,1|2 + |v1,3|2) + (|v3,2|2 + |v2,3|2) =1

3+

1

3+

1

3= 1

(1.2)

where |x| is the norm of the vector x.

To control the signaling dimensions used for decoding at the relay, we can only

change the precoding vectors, as the channel matrix is determined by the system

environment. The precoding vectors are designed and selected in such a way that the

product of the channel gain matrix and the precoding vector of the mutual signals

are the same as shown in Fig. 1.4. Hence the requirements for the precoding vectors

to implement SA are shown below:D12 = Hr,1 · v2,1 = Hr,2 · v1,2

D13 = Hr,1 · v3,1 = Hr,3 · v1,3

D23 = Hr,2 · v3,2 = Hr,3 · v2,3

(1.3)

After the SA requirements are met by processing messages with proper precoding

vectors, the signal received at the relay is:

yr = (Hr,1 · v2,1 · s2,1 +Hr,2 · v1,2 · s1,2)

+ (Hr,1 · v3,1 · s3,1 +Hr,3 · v1,3 · s1,3)

+ (Hr,2 · v3,2 · s3,2 +Hr,3 · v2,3 · s2,3) + nr

= D12 · s12 +D13 · s13 +D23 · s23 + nr

= D · [s12, s13, s23]H + nr

(1.4)

where D is of size 3 × 3 and is the concatenation of D12, D13 and D23, and sij =

si,j + sj,i. It is worth observing that if si,j = ±1 then sij = ±2 or 0.

At this point, the relay can use the zero-forcing (ZF) approach by multiplying

the received signal with the inverse matrix of D [16]–[18], or minimum mean square

error (MMSE) [19] or maximum likelihood (ML) decoder to recover each aligned

signal [20]. When BPSK is the modulation scheme for users, the decoded signals are

ternary symbols at the relay. To simplify decoding for users, the relay encodes the

9

decoded symbols. Specifically, ±2 is encoded as 0, and 0 is encoded as 1 [5], so that

the bits that the relay broadcasts are (b2,1 ⊕ b1,2), (b3,1 ⊕ b1,3), (b2,3 ⊕ b3,2), and users

can easily use XOR operation to obtain the desired messages.

1.3.2 BC Phase

In the BC phase, the relay broadcasts aligned signals to users, and users only

decode signals from the dimensions that carry the desired signals. To allow users

to have the ability to recover signals of interest, a pre-processing of signals with

precoding vectors at the relay must be deployed. There are two methods to obtain

the precoding vectors in the BC phase. One way is to use nulling precoding vectors

that exploit the effects of the channel matrix between the relay and the user, so that

destination terminals will not receive the signal processed by this nulling vector [14].

Another approach is based on the reciprocity assumption that Hr,i = HHi,r.

In the first method, the nulling vectors for each aligned symbol are denoted by

U12, U13 and U23, each of size 3× 1. The signal transmitted from the relay xr of size

3× 1 can be represented as:

xr = U12 · s12 + U13 · s13 + U23 · s23

= U · [s12, s13, s23]H(1.5)

where U of size 3 × 3 is the concatenation of U12, U13 and U23. To prevent users

receiving undesired aligned signals (e.g., user 1 is not interested in the aligned symbol

s23), the nulling vectors need to satisfy the following equations:H1,r · U23 = 0

H2,r · U13 = 0

H3,r · U12 = 0

(1.6)

Each equation in (1.6) is equal to three unknowns and three linear equations, so

there is a solution for each nulling vector. Then we take user 1 for instance and it

will receive:

y1,r = H1,r · U · [s12, s13, s23]H

= H1,r · U12 · s12 +H1,r · U13 · s13 +H1,r · U23 · s23

= H1,r · [U12 · s12 + U13 · s12]

(1.7)

10

In the same way, user 2 and user 3 independently receive:

y2,r = H2,r · [U12 · s12 + U23 · s23] (1.8)

y3,r = H3,r · [U13 · s13 + U23 · s23] (1.9)

To show how users obtain the desired information bits, we take user 1 as an example.

After processing through the channel between the relay and user, the received signal

at user 1 is:

y1,r = H1,r · U1 · [s12, s13]H (1.10)

where U1 is the concatenation of [U12 | U13

]. Then user i can decode aligned symbols

using matrix U1 inversion and obtain the desired message from other users by XOR-

ing its known bits with the decoded bits:

bi,j = bj,i ⊕(bi,j ⊕ bj,i

)(1.11)

1.4 Bit Loading Algorithm

In this section, we review, in the context of multicarrier transmissions, the con-

cepts of adaptive modulation schemes with modulation level selection and power

allocation. The reason for this is that in this thesis we deploy these concepts in the

spatial domain of signalling dimensions which constitute parallel channels in the same

way that frequency subcarriers do in multicarrier systems.

At the receiver, because of frequency selective channel characteristics, the instan-

taneous SNR can vary for different subcarriers. Assuming fixed modulation levels and

power across all subcarriers, the high instantaneous SNR at the specific subcarrier

would result in very low BER, and the subcarrier’s low instantaneous SNR would

result in high BER. In practical system, usually there is a targeted BER, indepen-

dent of subcarrier indices. Moreover, transmitting with a fixed power and modulation

level requires the overprovisioning of power, and this results in inefficient use of mod-

ulation level (bandwidth) on some subcarriers. Hence to obtain the acceptable BER

on all subcarriers, the amount of energy and the number of bits allocated to each

subchannel should be optimized according to the parallel channel gains (SNRs).

Bit loading algorithms allocate transmit power to each subcarrier based on their

channel conditions and then distribute bits to subcarriers to achieve throughput

11

improvement. There are two major types of bit loading algorithms: (i) minimizes

power subject to a data rate and BER constraints, which is called Margin-Adaptive

(MA) [21]–[23] and (ii) maximizes spectral efficiency with power and BER constraints,

which is called Rate-Adaptive (RA) [24]–[26]. Although the two types have different

constraints and objectives, they share the same principle of power efficiency.

In this thesis, in Chapter 4, we attempt to maximize the data rate and in the

following sections we present two known algorithms in the literature to reach the

maximum throughput with the constraint of fixed total power. First, we describe the

water-filling algorithm which is the optimal power allocation strategy that achieves

the capacity of parallel channels. Then we explain the operation of a practically

implementable algorithm to maximize throughput with a discrete alphabet signalling.

1.4.1 Water-filling Algorithm

As defined in information theory by Shannon, channel capacity is the tight up-

per bound of the maximum data rate that a channel can reliably transmit without

constraints on delay or coding techniques [27]. When the transmitter and the re-

ceiver know CSI through channel estimation and transmitter feedback, the optimum

power distribution strategy among subchannels to achieve the channel capacity is

water-filling.

Figure 1.5: Power allocation in the water-filling algorithm.

Fig. 1.5 illustrates the idea of the water-filling algorithm by an example of a

system with N = 7 subchannels. In the figure,√λi (i = (1, 2, ..., 7)) is the channel

gain of the subchannel i and xi is the power allocated to the subchannel i (the shadow

parts). Here, we assume the same noise power for all subcarriers. As presented in

the figure, power is distributed to the subchannels like pouring water over a surface.

12

The allocation of power is based on the inverse of the channel power gain (λi), which

guarantees that the subchannels with better conditions have more power compared to

other subchannels, but as the total amount of power is limited, sometimes the worst

subchannels do not have any power and are not used for transmitting [28]–[30].

The water-filling algorithm is to recursively solve N + 1 unknowns (xi and K)

with N + 1 equations as given by:λ−1i + xi = K (i = (1, 2, ..., N))∑Ni=1 xi = P

(1.12)

until gaining an acceptable solution. We can obtain one set of solutions from (1.12),

but when the solutions include a negative xi, it indicates that at least one subchannel

is not acceptable and should not be used. If the negative power result occurs, we set

the lowest xi in (1.12) to zero and omit the equation with λ−1i , and then recalculate the

equation set. The process is conducted iteratively until all outcomes are non-negative,

and then bits are allocated according to the channel conditions and allocated power.

However, the solutions of the water-filling algorithm are of theoretical value to

achieve capacity, and the allocated bits per symbol can be any real numbers, which is

impractical to implement in the modulation schemes alone [31]. Usually, the discrete

modulation schemes such as QAMs and phase shift keying (PSK) are used in adaptive

modulations, and for these discrete input alphabets, alternative bit loading algorithms

are considered [32].

1.4.2 Discrete Bit Loading Algorithm

Many researchers have investigated the discrete bit loading algorithms, and there

are three approaches seen as classic solutions: (i) the Hughes-Hartog algorithm which

is based on greedy optimization method [33]; (ii) Chow’s algorithm that distributes

bits by approximately rounding the water-filling results [34] and (iii) the Fischer-

Huber algorithm which provides solutions for systems with the constraints of total

energy and data rates [21]. As we aim for throughput improvement, the Fischer-

Huber algorithm of the MA type is not suitable; Chow’s algorithm is simpler than

the Hughes-Hartog algorithm, but it gives the worst performance out of the three

methods [35]. Hence, we select the Hughes-Hartog algorithm that offers the optimal

13

discrete bit loading results.

The Hughes-Hartog algorithm iteratively assigns one bit at a time to the subcarrier

that requires the minimum incremental amount of power to transmit one additional

bit at a targeted BER until the total power constraint is reached. Specifically, the

bit and power allocation proceeds in the following steps: (1) the power needed for

transmitting one more bit at a target BER is calculated for each subcarrier based on

their subchannel conditions; (2) check if the unallocated power is greater than the

minimum power obtained in the previous step; (3) if it is, one bit is assigned to the

subcarrier with the minimum required power and the leftover power decreases by the

amount of the power needed for this additional bit in this time, then repeat from step

(1); if it is not, it means that the leftover power cannot afford to transmit one more

bit, and the leftover power is distributed evenly to all subcarriers to enhance their

performances, and then power and bit allocations are finished. It should be noted that

the increment is one bit for the modulation schemes applied in the Hughes-Hartog

algorithm as it assigns power based on comparing the situations of one additional bit

transmission for subcarriers.

1.5 Thesis Objectives

This thesis designs a signal alignment scheme that exploits orthogonality of spatial

signalling dimensions in the MIMO Y channel to overcome limitations on controlling

the minimum distance between signaling points representing network coded messages.

This is accomplished by adding one redundant antenna at every user and calculat-

ing the corresponding precoding vectors to obtain the flexibility in the selection of

orthogonal signaling dimensions as well as to simplify the decoding at the relay. We

initially work with BPSK signaling and then we introduce procedures to accommo-

date higher modulation levels offering the advantage of higher bandwidth utilization.

Once we have a baseline for working with different modulation levels, more advanced

power allocation and bit loading are proposed to increase the data throughput in the

MIMO Y channel.

In the first part, the thesis explores the situation where the signal dimensions of

MIMO Y channels are designed to be orthogonal. It brings two advantages: (1) easy

decoding process at the relay without noise enhancement present when conventional

14

zero-forcing decoding algorithms are deployed and (2) improved BER performance

for user pairs though these improvements are unbalanced for different user pairs. For

BPSK signaling, we even the performances via power allocation to distribute energy

among the users based on their channel conditions. In addition, iterative optimization

of the signaling dimensions and time scheduling of transmissions according to channel

states are employed to further improve the BER performance.

In the second part, this thesis investigates the deployment of higher level mod-

ulations at users sides and the corresponding decoding at the relay with a unique

bit mapping strategy. Bit loading and power allocation strategies are developed to

further increase the MIMO Y channel throughput.

The original MIMO Y channel deploys the SA technique which combines two

mutual symbols in one dimension in the signaling space. This leads to significant

improvement in throughput, but the BER performance is not very good. This is

because sometimes the calculated signalling (spatial) dimensions for aligned symbols

result in the very low minimum distance between the symbols, and this affects symbol

error rate (SER) performance. The motivation for our work was that by ensuring

orthogonality of spatial dimensions, we will not only simplify the decoding process

but we will also control better the minimum distance between the signaling points

and thus reduce SER. Therefore, we attempt to add flexibility in the selection of

signaling dimensions to be able to work with orthogonal dimensions. By adding

redundant antennas and using iterative search, the signaling dimensions with best

effective channel gain for the aligned symbols are selected. In fact, the more redundant

antennas are used, the more diversity gain is introduced; and the more iterations are

applied, the more opportunity there is to obtain the better signaling dimension for

better BER performance. However, in reality, applications require simplicity and

efficiency, and adding many redundant antennas and exhaustive searches for better

signaling dimensions fail these requirements. These are the reasons why sometimes

in our work we were pursuing suboptimum solutions using algorithms with lower

computational complexity.

One of the main contribution in this thesis is the constellation mapping of QAM

symbols in the MIMO Y channel. As presented in the thesis, the constellation of

received aligned symbols at the relay is much more complex than the symbols sent

15

from users. To reduce the decoding calculations for all users, the relay can conduct

decoding for users and then re-encode the symbols to a smaller (original) constellation.

The generic ML decoding implemented through minimum distance decoding is

the optimal method which minimizes the symbol error rate, but its application is

computationally intensive in the MIMO Y channel at the relay. On the other hand,

ZF decoding is much easier but it suffers from noise enhancement, which sometimes

has negative effects on BER. The proposed scheme with only one extra antenna at

each user side builds on the fact that this antenna configuration offers infinite possible

combinations of the signal dimension selections. To efficiently decide the aligned

signal dimensions, we introduce the orthogonality constraint to the selection of the

dimensions. The proposed scheme with orthogonal signaling provides a simple way to

perform the decoding in MIMO Y channels through slicing of coordinates along the

orthogonal dimensions and is equivalent to ML decoding with hyper-cube signaling as

shown in Fig. 1.6. In hyper-cube signaling, a received signal is in a 3-dimensional (3D)

space that can be represented by three orthogonal coordinate axes, and this signal

can be decoded into three bits. Each bit is decided by the projection of the signal

on one axis: according to the distance of the projected coordinate and two possible

points on that dimension, the closest point is chosen to be the transmitted symbol

and one bit is decoded based on the selected symbol. The difference is that in hyper-

cube signaling, each dimension has two levels, while in the proposed scheme, each

signaling dimension has multiple levels that are determined by superposed symbols

and their modulation type used at users.

Figure 1.6: The Hyper-cube signaling.

16

1.6 Thesis Organization

The remainder of this thesis is organized as follows:

In Chapter 2, we propose the design of orthogonal signal dimensions for MIMO Y

channels which simplifies the ML decoding process. Computationally efficient method

is developed to calculate the precoding vectors; however, the method introduces un-

balanced BER performance among different users. To alleviate the problem and

further improve BER performance, different modifications in the original orthogonal

signal alignment scheme are proposed. As demonstrated through simulations, all pro-

posed schemes with BPSK signaling show improvement in the BER performance over

the original MIMO Y channel scheme.

In Chapter 3, we first present the QAM constellation mapping in the MIMO Y

channel to improve the system throughput. This is done through the bit mapping at

the user side and the extended constellation decoding at the relay. Then we develop

the bit loading algorithm to improve the utilization of bandwidth and power in the

system. The modulation selected in bit loading for each user pair depends on the

instantaneous SNR of the aligned symbol links. Hence, when power allocation is

done using bit loading, it maximizes the system throughput.

Chapter 4 presents the contributions of this thesis and the potential for future

investigations.

Appendix A derives the BER performance of the superposed constellations at the

relay which is used in Chapter 3.

Chapter 2

Orthogonal Signal Space Alignment for MIMO Y Channels

The antenna configuration in the original MIMO Y channel system imposes lim-

itations on controlling the minimum distance between signaling points representing

network coded messages, which affects BER performance. To mitigate this problem,

we add one redundant antenna at every user and design the corresponding precoding

vectors to obtain flexibility in the selection of orthogonal signaling dimensions and

to simplify decoding at the relay. The design of precoding vectors in the proposed

scheme improves the total BER performance despite the BER is not balanced among

the three users. To even the performance, the power allocation and iterative search

can be applied to distribute energy among the users based on their channel condi-

tions. In addition, the optimization of the signaling dimension orientation and time

scheduling are deployed to further improve the BER performance.

This chapter is organized as follows. First, Section 2.1 presents the previous work

focused on improving BER performance for MIMO Y channels. Section 2.2 demon-

strates the proposed system model. Next, Section 2.3 describes the advantages of

applying orthogonality, and the constraints for introducing orthogonality into MIMO

Y channel, as well as a simple method to conducting the calculations of precoding

vectors. Section 2.4 presents further modifications for the proposed scheme which lead

to better BER performance. Finally, different results are summarized and compared

in Section 2.5.

2.1 Related Literature Evaluation

In order to accommodate the two mutual messages into one dimension in the

signal space, multiple antennas are required. However, with the minimum antenna

configuration (Nu = 2 and Nr = 3), the BER performance of MIMO Y channel is

not ideal even with large transmit SNR, because sometimes the spatial dimensions

formed by the precoding vectors and channel matrices do not make the best use of the

17

18

aligned signal power. Moreover, the precoding vectors generated from the minimum

configuration are unique and we cannot avoid the situations when the undesired

spatial dimensions appear. Hence, some designs have been studied applying more

antennas to obtain more degrees of freedom for improving BER performance.

In [16], using redundant antennas, 2Nu −Nr random iterations are used to gener-

ate different beamforming vectors, and the set of precoding vectors that can provide

the best BER performance is selected as the final ones for the system. This algorithm

shows improvement compared to the minimum antenna configuration scheme, but

the random selection in the signal space may miss the opportunities for better per-

formance. Another iterative beamforming optimization algorithm is proposed in [18]

based on orthogonal projections in the signaling space, which shows much improve-

ment in the BER performance. However, the beamforming vectors in this design are

formed and selected through very complex calculations, and because of the irregular-

ity that appeared in iterations, it is difficult to theoretically proof the convergence.

An algorithm with lower computational complexity called an iterative interference

alignment is presented in [19], where in MAC phase precoding vectors and combining

vectors which are used for decoding at the relay are iteratively optimized at transmit-

ters and receivers. This aims to minimize the effects of interference signals from the

desired signals. However, in this algorithm, in each iteration the precoding vectors

and combining vectors are both considered, and the number of iterations required

for convergence is large. Moreover, [36] proposes antenna selection among redundant

antennas at users and the relay, but it is not ideal for practical applications because

the level of BER improvement depends on the number of used redundant antennas.

2.2 System Model

We use the MIMO Y channel model as in Fig. 2.1 where each user and the relay

have three antennas. In our derivation of orthogonal signaling in this case, we follow

in this chapter the notations introduced in Section 1.3. The only difference between

the proposed system model and the original one is the number of antennas at users,

which will change the channel matrices’ size to 3× 3 and the precoding vectors’ size

to 3× 1.

19

Figure 2.1: The proposed 3-user MIMO Y channel model.

In the original MIMO Y channel, each user is equipped with two antennas, re-

sulting in Hr,i being of size 3 × 2 and each precoding vector of size 2 × 1. For each

constraint in (1.3), there are four unknowns representing entries in two precoding

vectors and three linear equations, which implies a unique solution to the direction

of one pair of precoding vectors except for a parameter of scaling. Hence, the three

signaling dimensions determined in the original MIMO Y channel are fixed. After

the dimensions are obtained, we also need to impose the limits on the power of the

precoding vectors (|vj,i|2 = vHj,i · vj,i) to meet the power constraint in (1.2). This

power limit adds a constraint to the scaling parameters of precoding vectors, which

will partially influence the effective channel gain. However, because of the Rayleigh

fading channels, the effective gains |D12|, |D13| and |D23| are different and varying,

the distance between the closest aligned symbols in 3D signaling space at the relay

may be large for separating some aligned symbols and small for others. The average

BER performance is eventually determined by the minimum distance between the

aligned symbols which cannot be controlled in the original MIMO Y channel scheme.

The problem is visualized in Fig. 2.2, when the relay receives signals with the

ternary symbol coordinates for the case where the terminals are using BPSK. Actually,

there are 33 = 27 possible superposed symbols of the three aligned symbols received

at the relay as illustrated in Fig. 2.3. But in Fig. 2.2 we only show their coordinates

in the similar way as in quadrature phase shift keying (QPSK), where quaternary

symbols can be scrutinized through the coordinates of BPSK along the in-phase

20

and the quadrature components (corresponding to D12, D13 and D23). The major

difference here is that while for QPSK, the in-phase and quadrature directions are

orthogonal, in the original MIMO Y channel SA, the directions D12, D13 and D23

are not. Another difference is that the norms |D12|, |D13| and |D23| are normally

not equal, and the minimum distances for different signal pairs are random and can

be very small occasionally. However, when averaging the BER over the Rayleigh

channels, the BER performance is always determined by the minimum distance or

the worst scenarios. This means that in a MIMO Y channel, we would have larger

minimum distance and effective channel gains if we could utilize directions well.

Figure 2.2: Possible coordinates along D12, D13 and D23 for received symbols at therelay in 3D.

Figure 2.3: The possible signal received at the relay through the signal summationover the air.

21

In the MIMO Y channel model adopted in this thesis, each terminal and the relay

are equipped with three antennas, so that Hr,i and precoding vectors vl,i and vp,i are of

size 3×3 and 3×1, respectively. There are six unknowns (representing entries in two

precoding vectors) and three linear equations for each constraint in (1.3), resulting

in the solutions for a pair of precoding vectors being in the 3D space. Therefore,

the choices for the aligned dimensions D12, D13 and D23 (determined by precoding

vectors) are much more flexible than in the original MIMO Y channel. We exploit

this potential advantage in the following section.

After the relay receives signals, a decoding process is conducted normally by ZF,

MMSE or ML methods to obtain the aligned symbols, and the decoded aligned sym-

bols are remapped to BPSK again for simple decoding for users. Next, a new set of

precoding/nulling vectors are applied to the aligned symbols before sending to users.

Each user i will only receive two desired aligned symbols sip and sil from two of the

three dimensions that are generated in the BC phase. Then for each received symbol

sij, user i has the knowledge of its transmitted symbol sj,i and can use XOR oper-

ation to obtain the desired message si,j. In this thesis, we focus on the MAC phase

because, as demonstrated in [16], the precoding vectors from MAC phase can also be

calculated to obtain the precoding vectors in the BC phase. On the other hand, we

can also apply the antenna selection to benefit from receiver selection diversity in the

BC phase. In this way, two out of three antennas are chosen at each user based on

channel conditions to form the minimum configuration of MIMO Y channels and the

nulling vectors are used at the relay.

2.3 Orthogonality Application

2.3.1 ML Receiver

In most publications on the MIMO Y channel, the relay obtains the network-

coded version of paired messages by using the ZF, MMSE or ML decoders. In the

MAC phase, we decide on the three ternary aligned symbols s12, s13 and s23 of levels

±2, 0. In this case, the ML decoding is equivalent to the minimum distance decoding

of 33 = 27 ideal symbols visualized in Fig. 2.3. These points are located in 3D

space with positions determined by coordinates (±2, 0) · |D12|, (±2, 0) · |D13| and(±2, 0) · |D23|. Obtaining the received signal coordinates and using Voronoi regions

22

for ML decoding do not work in the original MIMO Y channel because the decision

rules are very complex and one needs to evaluate 27 Euclidean distances in 3D for

each packet. In the ZF approach, the recovery of the signal of interest is obtained

rather simply by inverting the matrix [D12|D13|D23] and multiplying the received

vector, but this method suffers from noise enhancement. In general, when compared

to ZF decoding, ML gives better performance (13dB gain at the BER of 10−3) but

requires higher computational complexity [20]. This motivates the investigation in

this thesis where we attempt to work with a simpler implementation of the ML

receiver by sacrificing some performance loss (in terms of BER) as a result of using

a constellation of points similar to hyper-cube signaling and a higher number of

antennas. Specifically, by imposing the orthogonality requirements, we simplify the

ML decoding at the relay to just projecting the received signal and slicing them

along three orthogonal directions D12, D13 and D23. In 2D, this corresponds to

interpreting quadrature amplitude modulation (QAM) or QPSK decoding as pulse

amplitude modulation (PAM) decoding along the orthogonal in-phase and quadrature

directions. In 3D, this approach is equivalent to decoding the hyper-cube signal as in

Fig. 1.6.

2.3.2 Orthogonality Constraints

As presented in the previous section, in the underlying system model, the SA

requirement for each signaling dimension constraint in (1.3) results in the solutions

to a pair of precoding vectors being in a 3D space. Because any pair of precoding

vectors in a 3D space defined by (1.3) can satisfy the SA requirement, we will uti-

lize this freedom by imposing extra conditions on precoding vectors (or equivalently

signaling dimensions). This is with the purpose to select precoding vectors that offer

benefits in terms of improved BER compared to ZF and to simplify the decoding pro-

cess compared to ML. Specifically, the proposed orthogonal signal alignment scheme

constrains the signaling dimensions to satisfy the following orthogonality conditions:DH

12 ·D13 = 0

DH12 ·D23 = 0

DH13 ·D23 = 0

(2.1)

23

where {·}H means Hermitian transpose. These are the additional constraints on

precoding vectors on top of those in (1.3).

2.3.3 Calculations of Precoding Vectors

We assume that the total transmit power PT = 1 is as in (1.2). To meet this

standard condition in the system under study, we work with the additional assumption

that the power for user i and user j to transmit their mutual symbols si,j and sj,i that

become an aligned symbol is constrained to 13, i.e., vHi,j · vi,j + vHj,i · vj,i = 1

3[20]. In this

way, we will analyze from the perspective of the links between users for simplicity

rather than users as each user accounts for two links which would be more complex

to discuss results.

Our objective is to calculate the precoding vectors (for each information symbol)

that will satisfy (i) the SA conditions in (1.3); (ii) signaling dimensions orthogonality

in (2.1) and (iii) the power limit (1.2). There is a total of six precoding vectors,

and each precoding vector is a matrix of 3 × 1, hence there are 18 scalar unknown

values. The nine linear equations in (1.3) and the three quadratic equations from (2.1)

compose a system of equations that will still give ample opportunities for choosing

precoding vectors. To reduce the complexity of solving these equations, we handle

the equation sets in a specialized sequence of steps involving linear matrix operations

presented next.

Before proceeding for the proposed SA, we rearrange our mathematical represen-

tations into more convenient forms. In particular, we substitute (Hr,2 ·v1,2), (Hr,3 ·v1,3)and (Hr,3 ·v2,3) for D12, D13 and D23, respectively, which gives the equivalent require-

ment to (2.1) as: (Hr,2 · v1,2)H · (Hr,3 · v1,3) = 0

(Hr,2 · v1,2)H · (Hr,3 · v2,3) = 0

(Hr,3 · v1,3)H · (Hr,3 · v2,3) = 0

(2.2)

We also transform the first equation in (1.3) into:[ v2,1v1,2

]= null[Hr,1 −Hr,2] (2.3)

where null[A] represents the null space of matrix A. [Hr,1 −Hr,2] is a 3 × 6 matrix

24

which with high likelihood has a full rank, hence the null space of this matrix has

6 − 3 = 3 dimensions and each dimension is represented by a vector of size 6 × 1.

These three dimensions are the orthogonal basis obtained from the singular value

decomposition for the null space of [Hr,1 −Hr,2], and the vector[ v2,1

v1,2

]can be equal

to any linear combination of these basis. That is to say, any vector in this null space

meets the requirement for v2,1 and v1,2 to align the mutual symbols. It should be

noted that there are three parameters for the three basis, which actually determine

the aligned dimension. At this point, we cannot decide which vector in this null

space is the best choice because in our proposed algorithm, the first selected aligned

dimension D12 will have influence on the other precoding vectors for D13 and D23.

Hence, at present, three random parameters are chosen for these basis and we can

obtain a 6 × 1 matrix that decides the direction of the precoding vectors v2,1 and

v1,2. Then we consider the power limitation by normalizing the basis parameters to

let[ v2,1

v1,2

]H ·[ v2,1

v1,2

]= 1

3, meaning that the power sum of v2,1 and v1,2, or the sum of

the transmit power for s2,1 and s1,2, equals to13. After v2,1 and v1,2 are decided, D12

is obtained as D12 = Hr,2 · v1,2.

Once we determine D12 as well as v1,2, we can find D13 and D23 with the first two

equations in (2.1). Because we are more interested in precoding vectors defining D13

and D23, from the equivalent orthogonality requirements in the first two equations

in (2.2) we calculate: v1,3 = null(vH1,2 ·HHr,2 ·Hr,3)

v2,3 = null(vH1,2 ·HHr,2 ·Hr,3)

(2.4)

which shows the precoding vectors v1,3 and v2,3 are in the same null space. For each

equation in (2.4), the left side is a vector of size 3× 1 meaning 3 unknowns, and the

right side is a null space of a 1 × 3 vector meaning one linear equation, so the null

space would have 3−1 = 2 dimensions and each dimension is represented by a matrix

of 3× 1. This indicates the null space is a plane where the precoding vectors v1,3 and

v2,3 would be. When deciding on v1,3 and v2,3, we also need to meet orthogonality

requirement between D13 and D23 as illustrated in the last equation in (2.2). To this

end, we use vectors x and y, each of size 3× 1, representing the two basis of this null

space and we use parameters a1 and b1 for representing v1,3 with the basis, and a2

25

and b2 to determine v2,3 with the basis, i.e., v1,3 = a1 ·x+b1 ·y and v2,3 = a2 ·x+b2 ·y.Now we substitute these representations of v1,3 and v2,3 to the last equation in (2.2)

and obtain the condition for a1, b1, a2, b2:[a1b1

]H · [x y]H · [Hr,3]H · [Hr,3] · [x y] ·

[a2b2

]= 0 (2.5)

So far, we cannot directly decide the best parameters a1, b1, a2 and b2 defining v1,3

and v2,3. For ease of calculations, we first choose two random values for a1 and b1 for

determining the direction of v1,3, and then use the second equation in (1.3) to get the

null space as: [ v3,1v1,3

]= null[Hr,1 −Hr,3] (2.6)

Although the basis for this null space has three dimensions, we can have the three

parameters for the basis with the knowledge of the determined direction of v1,3 and

hence obtain v3,1. At last, we normalize the 6 × 1 vector in this null space to 13to

meet the condition for power. With this, D13 is determined.

After a1 and b1 are selected, (2.5) leaves two unknown parameters a2 and b2

defining v2,3 which indicates v2,3 is a line in the space, so the direction of the precoding

vector v2,3 is fixed. Next, we transform the last equation in (1.3) to:

[ v3,2v2,3

]= null[Hr,2 −Hr,3] (2.7)

Still, the basis for this null space has three dimensions, but knowing the direction

of v2,3, the parameters for the basis in this null space can be decided and then the

direction of v3,2 is also obtained. Finally, we normalize the parameters to maintain

that |v2,3|2 + |v3,2|2 = 13. Till this point, all the precoding vectors and the aligned

dimensions D12, D13 and D23 are determined with the mutual orthogonality property.

2.3.4 Performance

In this section, we present the BER performance simulation results in the MAC

phase for our proposed scheme and the original MIMO Y channel scheme for com-

parison. The performance of the schemes without orthogonality are obtained with

ZF decoding as it is a widely used method in other research. The transmit SNR is

defined here as PT/σ2, where σ2 represents the variance of the noise. Hence, despite

26

how many antennas users are equipped with, the total transmit power in each system

remains the same for fair BER performance comparisons. All the links are assumed

to be quasi-static Rayleigh fading channels. The uncoded BPSK is adopted as the

modulation format, so that the network coding in the MIMO Y channel is the bit-wise

XOR.

The BER performance of the original and the proposed schemes is shown in

Fig. 2.4. For the original scheme, three user pairs present the same performance,

however, for the proposed scheme the figure shows different results for the three user

pairs. user pair23 appears to have a very close result to the one in the original MIMO

Y channel scheme, while for user pair13 there is an improvement of 5dB at BER

10−3, and for user pair12 it is even 18dB better than in the original MIMO Y chan-

nel scheme. Although the improvements do not distribute equally for the three user

pairs, the outcomes show progress in the proposed scheme.

SNR (dB)0 10 20 30 40

Bit E

rror R

ate

10-3

10-2

10-1

original schemeproposed scheme:pair12proposed scheme:pair13proposed scheme:pair23

Figure 2.4: The comparison between the BER performances of the original MIMO Ychannel scheme and the proposed scheme for different user pairs.

The reasons for unbalanced performance in the proposed scheme are as follows.

At the end of the last section, D12, D13 and D23 are calculated to meet the imposed

conditions for SA and orthogonality which may seem symmetrical. However, due to

the sequence of steps when solving the equation sets, the average powers of D12, D13

and D23 show difference: |D12|2 is the largest and then is |D13|2, and |D23|2 is the

27

minimum; and hence the performance of user pairs on different dimensions appears

to be unbalanced. The user pair12 on D12 has the best BER because the precoding

vectors for D12 is first decided and have a 3D space selection range, while user pair23

on D23 shows the worst BER for the precoding vectors of D23 are determined last

and only have one direction.

In addition, in Fig. 2.5, we present the throughput of the proposed scheme.

Throughput is defined here as the number of correctly received bits per time slot per

user pair. As the figure shows, the proposed scheme provides three different through-

put curves for the three user pairs, corresponding to their BER performance (better

BER performances lead to higher throughputs). In the original MIMO Y channel,

the three user pairs would have the same throughput similar to the user pair23 has in

the proposed scheme. When the BER performance is near 10−3 for each user pair, the

throughput for that pair is very close to one, which is the maximum that the BPSK

modulation can supply, and it demonstrates that in this scheme, the BER 10−3 would

be considered as an acceptable value for the system and will be used in the following

parts.

SNR(dB)0 10 20 30 40

Thro

ughp

ut

0.5

0.6

0.7

0.8

0.9

1

pair12pair13pair23

Figure 2.5: The throughput in the proposed scheme for different user pairs.

28

2.4 Modifications in Orthogonal Signaling

2.4.1 Power Allocation

We observe that the mutual symbols on D12 have the most improvement in BER

reduction over the conventional MIMO Y channel scheme, and the symbols on D23

have the minimal improvement. To let the performance be fair for each user pair, we

propose to apply power allocation to adjust the transmit power based on precoding

vectors depending on their channel characteristics.

As in Section 2.3.3, the sum power for transmitting mutual symbols is initially

fixed to 13, but when the aligned dimensions are obtained through the process in

Section 2.3.3, the mean values of the effective channel power gains (|D12|2, |D13|2

and |D23|2) are dissimilar: E {|D12|2} is the largest and E {|D23|2} is the smallest.

This is consistent with the BER performance of the three user pairs in Section 2.3.4.

To balance the outcomes, we attempt to retain the power equilibrium among the

three dimensions by adjusting the transmit power of users. We insert scaling factors

to the precoding vectors, and the representation of total transmit power is changed

from (1.2) to:

p12 ·(|v2,1|2+|v1,2|2)+p13 ·(|v1,3|2+|v3,1|2)+p23 ·(|v2,3|2+|v3,2|2) = 1 (2.8)

where, because of (1.3), p12, p13 and p23 are the power scaling parameters for D12, D13

and D23, respectively. In order to let all aligned symbols received at the relay have

the same norm for the effective channel gains (which for orthogonal signaling means

the same minimum distance) without modifying the total power limit, the following

relations are established: p12 · |D12|2 = p13 · |D13|2

p12 · |D12|2 = p23 · |D23|2

p12 + p13 + p23 = 3

(2.9)

when solving for p12, p13 and p23 which control improved effective channel gains. In

essence, the power scaling parameters p12, p13 and p23 redistribute the power from the

user pairs with better BER to compensate the user pair with the worst BER while

maintaining the same total power as in the original system.

29

The results are demonstrated in Fig. 2.6, where the three user pairs have matching

performance. This BER outcome is 4dB better than the original scheme. This is

because even though the result here is the average of the proposed three user pair

performances in Fig. 2.4 where one user pair’s BER performance is considerably

better than the one in the original scheme, the whole system is still under the major

influence of the worst link. Furthermore, comparing the BER results with the ones

in Fig. 2.4, it can be observed that the averaged performance is 4dB better than the

worst user pair but it is 15dB worse than the best user pair in the original orthogonal

signal alignment. Therefore, it seems inefficient to sacrifice a noteworthy gain from a

user pair with good performance to improve a little for another user pair. Again, it

is proven that for the BER performance the worst scenario has the most important

weighing factor for the total system.

SNR (dB)0 10 20 30 40

Bit E

rror R

ate

10-3

10-2

10-1

pair12pair13pair23

Figure 2.6: The BER performance of the proposed MIMO Y channel scheme withpower allocation.

2.4.2 Time Scheduling

Even though the power allocation can equalize the BER performance for the three

user pairs, the balanced result is dominated by the worst user pair’s performance.

Hence we propose to control the occurrences of the worst scenarios by imposing

limitations in time on terminals accessing the orthogonal spatial streams when the

30

channel conditions result in any of the effective channel power gains |D12|2, |D13|2

and |D23|2 below a predefined threshold. If the powers of signalling dimensions fall

below a certain predefined threshold, the users are withholding their transmissions of

mutual symbols until the channel conditions become more favorable. This precaution

approach decreases the amount of useful TSs as well as the network throughput but

would significantly improve the average BER performance by avoiding transmissions

when the channel conditions are not favorable. Through this method, the worst cases

are under control and have less effects on the overall BER performance. Here we

assume that when an effective channel power gain is below a threshold, three users

will all quit transmitting.

We have three ways to conduct the time scheduling, which will demonstrate the

relationship between the throughput and the BER performance. The first method is

that the three effective channel power gains are individually compared to the prede-

termined threshold and that users transmit messages only if all channel gains exceed

the threshold. The user pair13 and user pair23 have rather low mean effective channel

power gains, hence they will have a stronger possibility than user pair13 to trigger the

withdrawing process. However, any channel that does not satisfy the threshold will

put all three users to sleep mode, and in this way, the BER performances of the three

user pairs would still be different. For obtaining balanced results, the second method

is to combine time scheduling with power allocation method when time scheduling is

used first. Also, we can apply the power allocation first and then check if the evened

effective channel power gain exceeds the threshold, which is the third method.

Different values for the threshold can be defined according to system performance

requirements. For demonstration, we choose the threshold to be 0.0173 here, with

the normalized system transmit power constrained to one, to examine the effective

channel power gains. The simulation result for the first method is shown in Fig. 2.7

which illustrates the performance when only applying time scheduling. The BER

results of user pair13 and user pair23 are improved around 10dB but user pair13 has

only 1dB improvement over the original scheme. This outcome is in accordance with

the earlier analysis as the effective channel power gains of user pair13 and user pair23

have larger chances to be below the threshold, hence the time scheduling method

would efficiently control the worst cases for both the user pairs and improve their

31

performances. However, this improvement is realized at the expense of a 24.75%

reduction of the useful TSs for all users and the performances among user pairs are

still different.

SNR (dB)0 5 10 15 20 25 30

Bit E

rror R

ate

10-4

10-3

10-2

10-1

pair12pair13pair23

Unused Time Slots:24.75% (Th = 0.0173)

Figure 2.7: The BER performance of the proposed MIMO Y channel scheme withtime scheduling.

Fig. 2.8 presents the relationship between the predefined thresholds and the cor-

responding loss in throughput when applying time scheduling. As seen in the figure,

the relationship is linear.

Fig. 2.9 presents the throughput of the three user pairs when applying time

scheduling and corresponds to the results in Fig. 2.7. The throughput versus SNR

results is different for the three user pairs, same as it was the case for their BER

performance. It is obvious that the throughput are decreased as compared to Fig. 2.5

for the original orthogonal signaling and the maximum throughput in this case is only

about 0.75, a reduction of around 25% of the maximum throughput in Fig. 2.5. It

should be pointed out that this reduction ratio is almost the same as the ratio of

the unused TSs by conducting time scheduling. This is because when SNR is large

enough and errors seldom occur, the throughput is basically equal to the transmit

rate, but when a part of TSs are wasted without transmitting, the overall throughput

must be reduced.

32

Figure 2.8: The relationship between the thresholds and the corresponding TS lossratio in time scheduling.

SNR (dB)0 5 10 15 20 25 30

Thro

ughp

ut

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

pair12pair13pair23

Figure 2.9: The throughput of the proposed MIMO Y channel scheme with timescheduling.

It is worth noting that the percentage of unused TSs is the same for any transmit

SNR, since the predefined threshold used in the time scheduling is directly compared

to the channel condition. This is reasonable for a system where power is crucial: when

33

a worst channel condition occurs, much larger transmit power or SNR is needed for

offsetting the negative channel effects and achieving the same performance that a

little power can obtain when the channel presents average condition. Hence, when

power is of great importance, the system would not waste most of it to compensate

for the worst channel condition. On the other hand, when power is not a great

concern, the threshold should be compared to the instantaneous SNR where transmit

SNR and channel conditions are considered simultaneously, which will be analyzed in

Chapter 3.

Next we consider the performance of orthogonal signaling with power allocation

applied after time scheduling to have balanced results for the three user pairs. The

BER results for this shceme are demonstrated in Fig. 2.10. Two different predefined

thresholds 0.0173 and 0.0035 are set here for affording comparisons. With the thresh-

old 0.0173, the BER performances of the three users achieve BER 10−3 at 18.5dB with

a cost of a 24.7348% reduction of useful TSs, while with the threshold 0.0035, for the

same BER target, the SNR requirement is 23dB but only with a price of 5.2023% de-

crease of useful TSs. This outcome illustrates a tradeoff between the thresholds with

the lost percentages of useful TSs and the BER performance: the higher the threshold

is, which leads to the higher ratio of unused TSs, the better BER performance would

be.

Finally, we analyze the performance of the third method, where power allocation

is applied first and followed by time scheduling. For demonstration, we show the BER

results in Fig. 2.11 with two thresholds 0.01 and 0.0378 for this case. It can be seen

there that to get BER 10−3, the SNR required is 23dB and 18dB for thresholds at

0.01 and 0.0378 with the drop of 5.8676% and 24.5884% of used TSs, respectively.

If we compare Fig. 2.10 and Fig. 2.11, the two figures are almost identical with

respect to the associated unused TS ratios but not with respect to the thresholds. It

seems that there is a clear relationship between the BER performance and the ratio

of unused TSs, which is rather reasonable because unused TSs result in less errors

and better BER performance. Therefore, the tradeoff to be exercised is between the

useful TS reduction and the improved BER performance rather than the thresholds,

and we can adjust either one to obtain the desired performance of the other one.

34

SNR (dB)0 5 10 15 20 25 30

Bit E

rror R

ate

10-4

10-3

10-2

10-1

Unused Time Slots:24.7348% (Th = 0.0173)5.2023% (Th = 0.0035)

Threshold: 0.0173Threshold: 0.0035

Figure 2.10: The BER performance of the proposed MIMO Y channel scheme withtime scheduling applied first and then followed by power allocation.

SNR(dB)0 5 10 15 20 25 30

Bit E

rror R

ate

10-4

10-3

10-2

10-1

Unused Time Slots:5.6876% (Th=0.01)

24.5884% (Th=0.0378)

Threshold: 0.01Threshold: 0.0378

Figure 2.11: The BER performance of the proposed MIMO Y channel scheme withpower allocation applied first and then followed by time scheduling.

As for threshold values for methods involving time scheduling, the larger threshold

will lead to a larger ratio of lost TSs (i.e., TS not used for transmission). However,

the method that uses time scheduling first always requires smaller thresholds than the

method that applies power allocation first for achieving the same performance. For

35

instance, when the ratio is 25%, the threshold is 0.0173 for the second method and

0.0378 for the third method; when the ratio is 5%, the threshold is 0.0035 and 0.01

for the second and third methods, respectively. Hence, the sequence of applications

of methods have influence on the thresholds. This is because, on one hand, when

conducting time scheduling first, the threshold is used to judge every user pair’s

effective channel power gain. On the other hand, while using power allocation first,

unacceptable effective channel power gain can be compensated from the redistribution

of transmit power which results in the balanced effective channel power gain greater

than the old worst gain of a user pair, and this increases the threshold in the time

scheduling phase.

Fig. 2.12 presents the throughput comparison between the first method and the

third method (the second method can have the same throughput results and BER

performance with the same ratio of reduced TSs as the third method but with dif-

ferent thresholds). The figure shows the maximum throughput with the unused TSs

percentage as the parameter.

SNR(dB)0 5 10 15 20 25 30

Thro

ughp

ut

0.4

0.5

0.6

0.7

0.8

0.9

1

power allocationpower allocation+time scheduling(Th: 0.01, unused TS ratio:5.6876%)power allocation+time scheduling(Th: 0.0378, unused TS ratio:24.5884%)

Figure 2.12: The comparison of the throughput in the scheme using power alloca-tion and in the scheme using both power allocation & time scheduling with differentthresholds.

36

To extract more information, we calculate the scale of the change in throughput

caused by time scheduling using results from Fig. 2.12. The scales are shown in

Fig. 2.13, where the vertical axis presents the reduced ratio of the throughput with

the scheme applying time scheduling and power allocation to the one with only power

allocation. In Fig. 2.13, the two curves first decrease and then rise to the percentages

of unused TSs. The front drop means that the time scheduling can avoid bad TSs in

the low-SNR regime when the effective channel gains are usually not desired, which

diminishes the probability of occurrence of errors. However, when SNR is larger and

errors seldom appear, the lost TSs will set a constraint to the achievable throughput in

the system. This indicates that there is a minor relationship between the throughput

and the SNR.

SNR(dB)0 5 10 15 20 25 30

Rat

io o

f red

uced

thro

ughp

ut

0

0.05

0.1

0.15

0.2

0.25

with 24.5884% unused TSswith 5.6876% unused TSs

Figure 2.13: The ratio of reduced used TSs when applying time scheduling & powerallocation compared to the one when applying power allocation.

2.4.3 Iterative Search for Optimized Orientation

The time scheduling method trades TSs for improvement in BER performance,

and the results in Section 2.4.2 illustrate that combined with power allocation, the

time scheduling algorithm can offer 14dB and 19dB improvements for all user pairs

compared to the original MIMO Y channel, at the expense of 5.2% and 24.7% losses

37

of useful TSs, respectively. Moreover, we can further improve the BER performance

and decrease the amount of unused TSs by employing iterative search for deciding

the orientation of orthogonal signaling dimensions.

Because the unbalanced performance among three user pairs is caused by the pref-

erence given to user pairs in sequencing the steps to calculate precoding vectors, the

proposed algorithm in this section uses six iterations that are generated by different

sequences in solving the equation sets (1.3) and (2.1) for optimizing orientation of

orthogonal effective channel gain. For example, we can calculate D12 first and then

D13, or calculate D13 first and then D23, and there are total six permutations. Hence,

in the algorithm, six sets of precoding vectors are independently calculated by iter-

atively solving six different ordered equations. Because the bottleneck of the system

performance is the user pair with the worst result, we select the best set of precoding

vector solutions which can offer the best effect on the worst user pair’s performance.

The selected sequence is actually random for each TS due to Rayleigh fading channels,

and hence for a user pair the chances are equal to have the best, medium or the worst

performance as exploited in the proposed orthogonal scheme. This randomization

balances the three user pairs’ performance, which is even better than the ones using

power allocation that also intends to average the overall performance.

The simulation results with only iterative search for optimized orientation are

shown in Fig. 2.14, providing almost 10dB improvement compared with the original

MIMO Y channel. Also, the selection among six sets significantly reduces the level of

the worst scenarios so that the overall performance using iterations is better than the

one with power allocation as in Fig. 2.6. As the iterative search offers selections to

utilize better the available signal space, it can be combined with the time scheduling

to reduce the number of unused TSs. To compare the performance of this combined

scheme with the results in Section 2.4.2, the same parameters for the third method in

Section 2.4.2 are set for the simulations here and the outcomes are shown in Fig. 2.15.

At BER 10−3, the SNR is at 18dB and 21.25dB with thresholds 0.01 and 0.0378,

respectively, and both provide 1dB improvement compared to the results without

iterations in Fig. 2.11. What is more important is that the losses of used TSs in the

scheme with iterations are reduced from 24.6% to 10.0% and from 5.7% to 1.8%, which

38

SNR (dB)0 10 20 30 40

Bit E

rror R

ate

10-5

10-4

10-3

10-2

10-1pair12pair13pair23

Figure 2.14: The BER performance of the proposed scheme with six iterations.

SNR (dB)0 5 10 15 20 25 30

Bit E

rror R

ate

10-4

10-3

10-2

10-1

Unused Time Slots:1.7992% (Th=0.01)

10.0054% (Th=0.0378)

Threshold:0.01Threshold: 0.0378

Figure 2.15: The BER performance of the proposed scheme with different appliedmethods.

is around a 60% saving of useful TSs. Therefore, the iterative search for orientation

combined with time scheduling and power allocation can let more TSs be useful and

even further improve the BER performance and throughput.

39

2.5 Conclusion

To compare the performance of different schemes and modifications in one place,

we show in Fig. 2.16 the simulation results of balanced BER performance using

schemes as identified in the legend of the figure. As seen in the figure, the power allo-

cation alone gives 5dB improvement (at BER 10−3) compared to the original MIMO

Y channel scheme. Actually, the orthogonality provides different improvements for

user pairs and the most improved one is 18dB. However, with power allocation, the

user pair that has the best improvement redistributes much of its transmit power to

compensate the worst user pair for a little improvement, but still the worst perfor-

mance owns a larger weighting factor in the whole system outcome, finally leading to

an average of 5dB improvement in the averaged performance.

SNR (dB)0 10 20 30 40

Bit E

rror R

ate

10-4

10-3

10-2

10-1

originalpower allocationiterationspower allocation + time schedulingpower allocation + time scheduling + iterations

Figure 2.16: The BER performance of the proposed scheme with six iterations com-bined with power allocation and time scheduling.

For iterative seach only, the improvement is 10dB compared to the original scheme.

Because of Rayleigh fading, the channel conditions are random, and hence if more

selections are available and the best of them can be chosen, it would bring improve-

ment for the BER performance. This method is utilized in some research papers such

as in [16] and [18].

40

Next, when power allocation is applied with time scheduling, the worst cases are

under control where users do not transmit messages when the effective channel gains

are not favorable. This brings 18dB improvement for all users when compared with the

original scheme but at a cost of a 24.6% reduction of useful TSs. The most significant

property of MIMO Y channels is its high throughput where six packets are exchanged

among three users in two TSs. However, using time scheduling, a perceived amount

of throughput needs to be lost for BER performance improvement, and this would

discount this major advantage. Hence, to reduce the lost of throughput and further

improve the BER performance, iterative search is combined with time scheduling. In

Fig. 2.16, when power allocation, time scheduling and six iterations are combined

together, the BER performance is 19dB better than the original scheme and it only

causes a 10% reduction of unused TSs. Compared to the results in [16] and [18] with

the same antenna configuration as used in the proposed scheme and with the same

target BER 10−3, the best performance shown in Fig. 2.16 requires 9.5dB or 6dB less

than in [16] where three or seven exhaustive iterations were conducted, respectively.

[20] proposed a coding technique for the original MIMO Y channels and realized a

coding gain of 23dB with a cost of a 75% reduction in throughput, but the scheme

shown in this thesis only requires around 10% loss of throughput to achieve 19dB

gain.

The orthogonality introduced in this thesis provides a simple way to perform the

decoding in the MIMO Y channels without noise enhancement as in using ZF. How-

ever, it also introduces uneven BER performance for user pairs and this characteristic

can be used in certain systems where there is a major communication link requiring

better conditions than other links, as the improvement for the first user pair is tremen-

dous. On the other hand, the signal space in the proposed scheme is not completely

utilized as in the calculations for precoding vectors, several parameters are selected

for simplicity. Therefore, the algorithm can be further improved with more sophisti-

cated choices for those parameters and more restrictions for the aligned dimensions.

The modifications are built under the condition that the balanced BER performance

is required. Hence, even though it may be inefficient to apply power allocation, where

much energy of the best user pair is sacrificed to compensate the worst user pair with

a little improvement, this is achieved with low computational complexity. The six

41

iterations can also lead to the balanced performance for the three user pairs and even

result in more BER improvement and a lower reduction of throughput, but the cost

is the increased complexity of the algorithm.

Chapter 3

Bit Loading and Power Allocation

The previous chapter was focused on optimizing BER performance in MIMO Y

channels with orthogonal signalling when BPSK was deployed at the terminal nodes.

The current chapter applies the Gray-coded M-ary QAM in MIMO Y channels with

orthogonal signaling to increase the throughput over the one bit per time slot in the

MAC phase with BPSK for one user pair. While the M-ary QAM has been investi-

gated for PNC [37] [38], this is the first application of M-ary QAM in the context of

the MIMO Y channels with orthogonal spatial signaling. From the perspective of one

user pair data exchange, when orthogonal signaling is deployed, the encoding with

M-ary QAM is very similar to the conventional PNC with one spatial stream. How-

ever, in MIMO Y channel we deal with three spatial streams and new opportunities

arise. Specifically, to optimize bandwidth and power resources in PNC in varying

radio channels, data rate and power are adapted based on the channel quality. In

this chapter, we apply rate and power adaptation in MIMO Y channel in a similar

fashion as this was originally proposed for PNC but we do this across three spatial

(parallel) streams corresponding to three user pairs.

This chapter is organized as follows. Section 3.1 discusses general ideas behind

rate and power adaptations. Section 3.2 presents the system model where the MIMO

Y channels can be treated as three parallel two-way relay channels. Next, Section 3.3

demonstrates the required knowledge to proceed the bit loading so that modulation

levels are selected based on the effective channel gain. In Section 3.4, the sub-optimum

bit loading with power allocation is added to the scheme to distribute transmit power

and to select proper modulation types for each user pair for achieving the maximum

throughput in the system. Finally, the simulation results are presented and discussed

in Section 3.5.

42

43

3.1 Rate and Power Adaptations

In the previous chapter, the thresholds used in time scheduling in fading channels

with BPSK are based on effective channel power gains which are the square of the

absolute value of the multiplication of channel gains and precoding vectors, and not

based on the instantaneous received SNR. The instantaneous SNR is the multipli-

cation of the transmit SNR and the effective channel power gains, and it directly

determines the BER performance. In this chapter, the rate and power adaptation is

controlled by the instantaneous received SNR [39].

We show an example to demonstrate the difference between the power/thresholds

decided based on (i) the effective channel gains and (ii) the instantaneous SNR.

Assume a system (not necessarily with BPSK signaling) where the instantaneous

SNR is required be ten to achieve BER 10−3 in a AWGN channel. Because AWGN

channel coefficients are always one, the effective channel gains are always one, so that

the transmit SNR is the same as the instantaneous received SNR. Therefore, the SNR

and BER relationship in AWGN channels is the same as instantaneous SNR and BER

relationship for fading channels. If this system intends to achieve the same BER in

a Rayleigh fading channel, the required SNR is depended on the channel coefficients

which vary in a wide range. For example, if at a moment the channel condition is bad

and the channel coefficient is 0.1 (channel power gain is 0.01), the transmit SNR is

required to be 1000 to let the instantaneous SNR be ten to achieve the aimed BER;

if at one moment the channel coefficient of the Rayleigh fading channel is one (the

average channel gain in Rayleigh fading channels), the transmit SNR is required to

be 10 to achieve the aimed BER.

This example shows that when channel condition is bad, a significant amount of

energy are required to compensate the negative influence caused by this channel for

achieving the same performance that could only consume a little power when the

channel is in its average status. Therefore, on occasions where the power efficiency is

of great importance, users should decide whether to transmit according to the channel

conditions. However, when it is not the case, users should send messages based on

instantaneous SNR without consideration of the transmit SNR as long as the BER

performance target is reached.

44

When users choose to transmit messages when channel conditions are acceptable,

they can also determine the modulation level of symbols based on channel quality.

More bits or a symbol with higher modulation level can be transmitted on a rather

good channel and less bits or a symbol with lower modulation scheme can be trans-

mitted on a rather bad channel, without influencing the BER performance. This

is called bit loading. By using bit loading, the throughput of the transmission can

be improved under a certain BER requirement. When the effective channel gain is

sufficient for applying a higher modulation scheme to achieve a targeted BER, the

system will use a higher modulation to increase throughput.

When there are multiple channels in a system, the transmit power for symbols on

each channel can be redistributed to improve the whole system performance, e.g., the

total sum rate. This is the bit loading with power allocation. The optimal one that

reaches the capacity of the system with parallel channels is the water-filling algorithm.

It allocates more power to messages on a good channel to support higher modula-

tion, and stop sending messages in unacceptable bad channels. However, water-filling

is more like a theoretical algorithm for channel capacity calculations where the bit

assignment on any parallel channel can be any real number. This may be difficult

to realize without designing integrated coding and modulation and this thesis is only

considering the latter. There are also schemes like the Hughes-Hartogs algorithm

which distributes power for maximizing throughput with discrete information bits

determining feasible modulation scheme on parallel channels. The original Hughes-

Hartogs algorithm for multicarrier transmission iteratively chooses the subcarrier that

requires the least power to transmit one additional bit and allocates it with the exact

amount of power needed. This method assigns the power with the aim to apply the

highest modulation that the power can support under a certain BER requirement,

so that the throughput is maximized with total power and BER performance con-

straints. In our proposed scheme, the Hughes-Hartogs algorithm is adapted to our

conditions.

3.2 System Model

In this chapter, the system configuration is the same as the proposed one in

Chapter 2, and channels are also assumed to be Rayleigh fading. As derived in (1.4),

45

in the MAC phase, the received signal at the relay is:

yr = D12 · (s12) +D13 · (s13) +D23 · (s23) + nr (3.1)

Because of the orthogonality in (2.7), when the decoding process is conducted at the

relay, the Hermitian transposed effective channel gain Dm (m = {12, 13, 23}) can

directly multiply the mixed signal in (3.1) and one can decode the symbol sm for the

user pairm from the processed signal:

rm = DHm · yr = DH

m ·Dm · (sm) +DHm · nr (3.2)

Then, when decoding an aligned symbol sm, the instantaneous SNR for the channel

with direction Dm is:

SNRinstant,m =|Dm|2

σ2(3.3)

With this, the received signal in (3.1) can be represented in each orthogonal direction

carrying one user pair’s data as:

yr,m = Dm · (sm) + nr,m (3.4)

where the noise components nr,m are also orthogonal and independent so that we

could decode user pairs’ data individually. Hence, the system model can be treated

as three parallel two-way relay channels.

3.3 Bit Loading

In this thesis, we implement BPSK and Gray-coded M-QAM modulations in the

proposed MIMO Y channel, as M-QAM are widely used in rate-adaptive communica-

tion systems. Specifically, we work at the user side with the modulation level M set

either to 2, 4, 16 or 64 to limit the size of the expanded constellation at the relay (3,

9, 49 and 225, correspondingly). It should be noted that if ML decoding is directly

applied, the relay need to evaluate 33, 93, 493 and 2253 ideal points, correspondingly,

but with the proposed scheme with orthogonal signaling dimensions, the decoding

process is simplified to projecting the signal and slicing its coordinates along the

dimensions, despite modulation levels.

The reminder of this section presents the necessary background for conducting bit

loading and is organized as follows. Section 3.2.1 demonstrates the problem occurred

46

at the relay when conducting the encoding process with higher modulation levels and

the corresponding solutions. Section 3.2.2 summarizes the thresholds for different

modulations as they will decide the modulation level for each channel.

3.3.1 Constellation Mapping

In this section, we show that the constellation size at the relay is expanded and

then how to perform decoding and encoding at the relay, so that in the BC phase, users

could recover data of interest. We first demonstrate the details of the bit mapping

process in the BPSK and 4-QAM modulation, and then present the problem that

when using higher modulation schemes than the BPSK and the 4-QAM, the XOR

encoding at the relay causes ambiguity. Finally, a solution is derived to resolve this

ambiguity.

The constellation size at the relay is different from the applied modulation at

users, as the received symbol at the relay is a superposed symbol of two symbols

from two users. When the modulation level increases, the constellation size of the

received symbol increases much faster. For symbols using BPSK of±1, the superposed

symbols will have three levels: ±2 and 0. Moreover, the 4-QAM leads to the received

constellation of size 3×3 at the relay, because 4-QAM is a superposition of two BPSK

along the in-phase and quadrature components. Furthermore, for 16-QAM the size

of the received constellation at the relay is 49, and for 64-QAM the size is 225. Again

this is because the 2-D M-QAM, where M =√M ·

√M is a superposition of two 1-D

√M -ary PAMs along the in-phase and quadrature components. Generally, for square

M-QAM at the user side, there is (2√M − 1)2 aligned symbol at the relay on one

spatial signaling dimension. This is because the received at the relay superimposed

PAM in either in-phase or quadrature would be of size 2√M − 1 [39]. If the relay

was in the AF mode and it was resending symbols from the expanded constellations,

this would result in the inefficient use of power for the aligned symbols, and moreover

each user would have to decode two such aligned symbols. However, when the relay

is in the DF mode, it can decode the received signals and re-map the symbols into a

smaller constellation, which simplifies the decoding process for all users and results

in the better use of power in the BC phase.

47

Now we show the mapping processes for different modulations at the relay. For

BPSK, the mapping process is illustrated in Fig. 3.1 where the blue and red bits

represent the information bits from two users and there are a total of 22 bit combi-

nations, with two of them 01 and 10 resulting in the same aligned (superimposed)

symbol decoded as 0 in the constellation. We can see it from the figure that the

re-mapped constellation representing the XOR-ed version of user bits at the relay is

still BPSK.

Figure 3.1: The decoding and remapping for BPSK symbols.

For 4-QAM, its in-phase and quadrature signals both can be considered as BPSK

and hence the mapping for 4-QAM will be a 2-D BPSK mapping as shown in Fig. 3.2.

The black dots are the aligned symbols of the yellow dots that are the original 4-QAM

constellation. In each dimension, the yellow dots are at ±1, and the aligned symbols

are at ±2 and 0 as in BPSK. The black bits represent mapped bits which are also

the XOR-ed version of two users’ bits.

So far, it seems that the re-mapping could be conducted following XOR of the

original information bits. However, the XOR mapping can cause ambiguity when the

modulation is higher than BPSK and 4-QAM [40]. Taking one dimension of 16-QAM,

i.e., 4-PAM with two bits from one user as an example, Fig. 3.3 shows the ambiguity

when trying to remap aligned 16-QAM symbols in one constellation dimension follow-

ing the process as in BPSK. When the superposed constellation symbols are equal to

+2 or −2, the XOR results of the gray-coded original symbols from two users are not

unique. If the relay applies this mapping, users cannot decode their bits of interest.

48

Figure 3.2: The decoding and remapping for 4-QAM symbols.

Figure 3.3: The ambiguity occurring in decoding and remapping for one dimensionof 16-QAM symbols.

In order to avoid the ambiguity, we have to ensure that a mapping between a user’s

symbol and the aligned symbol is unique. In Fig. 3.3, the XOR result 01 representing

constellation ±4 and the XOR result 10 representing constellation 0, each has four

combinations. In either group of these four combinations, the four blue information

bits are different and the red bits are also different. This indicates that there is a

49

one-to-one relationship between the blue bits and the red bits under the XOR results

01 and 10 (which will allow recovery of data of interest at the user side in the BC

phase assuming the user remembers its own data). Consequently, for the two remain-

ing available mapping bits 00 and 11, each should also represent four combinations,

despite XOR results, with a condition that in the group of four combinations the blue

and the red information bits need to have one-to-one relationship. For instance, if the

mapping bits 00 represent constellation −6 to have the combination 0000, it cannot

represent −2 because there is a combination of 0011 where 00 is repeated with the

blue bits in the former combination 0000. One possible mapping that satisfies this

rule is shown in Fig. 3.4.

Figure 3.4: The decoding and remapping of 4-PAM symbols corresponding to 1-D in16-QAM signalling.

Table 3.1: The bit mapping for the combination of two 4-PAM symbols.

Mapped Bits 00 01 11 10Bit Combinations 0000 0001 0011 0010of Two Symbols 0110 0100 0101 0111Individually from 1111 1110 1100 1101

Two Users 1001 1011 1010 1000

In addition, corresponding to Fig. 3.4, the mappings of the combinations of two

original symbols are summed up in Table 3.1. It can be seen that under each group of

the mapped bits, if a symbol in blue or red is known, the other symbol can be decided.

For instance, an aligned symbol is remapped to bits 11, and this remapped symbol is

sent to users; if the blue user receives bits 11 and knows its former transmitted bits

are 11, it can decode the desired bits 00 from the red user according to Table 3.1.

This one-to-one mapping between two original symbols under a mapped symbol at

the relay provides a look-up table for users without ambiguity. XOR mapping can still

be used for simplicity where no uncertainty occurs. Wang et al. proposed a mapping

algorithm involving modulus [40], but our method is simpler and faster via a direct

mapping table for each modulation scheme.

50

3.3.2 Modulation Thresholds

In this section, we derive the thresholds for determining modulation levels for

each of the three spatial channels in our system model with orthogonal signaling. To

obtain the thresholds, we need to determine the targeted BER and the BER per-

formance of the superposed symbols received at the relay. Moreover, because the

instantaneous SNRinstant,m in fading channels has the same effect one the BER per-

formance as in AWGN channels, the BER performance of different constellations for

those superposed symbols in AWGN channels will be obtained and used to determine

modulation levels for the proposed scheme in Rayleigh fading channels.

In this thesis, we aim at the BER 10−3, and the corresponding thresholds when

switching between different modulations based on SNR can be determined from

Fig. 3.5, which shows the BER as a function of the received SNR (per superposed

symbol) for different modulation levels of the received symbols at the relay. The bit

SNR (dB)0 5 10 15 20 25 30

Bit E

rror R

ate

10-4

10-3

10-2

10-1

3 QAM (BPSK)9 QAM (4QAM)49 QAM (16QAM)225 QAM (64QAM)

Figure 3.5: The BER performance of the received symbols at the relay with differentmodulations in AWGN channels.

error rate performance for the superposed constellations is analyzed in Appendix A,

where we reach the conclusion that the bit error rate performance of the superposed

constellations at the relay is 3dB worse than that of the original constellation at the

user sides. Therefore, to obtain the plots in Fig. 3.5, we could use the readily available

51

expressions for the conventional QAM constellations used at the user sides with the

appropriate shifts [28]. The thresholds for the SNR at the relay to decide when to

switch between different original constellations at the user sides are summarized in

Table 3.2. As a side note, these thresholds are related to the SNR at the transmitter

which is used when analyzing the performance of our system in the simulation section.

Table 3.2: The modulations and the corresponding thresholds for achieving the aimedBER 10−3.

Modulations Thresholds RangeNO TX — 0 dB < SNR ≤ 9.9 dBBPSK 9.9 dB 9.9 dB < SNR ≤ 12.9 dB4-QAM 12.9 dB 12.9 dB < SNR ≤ 19.7 dB16-QAM 19.7 dB 19.7 dB < SNR ≤ 25.7 dB64-QAM 25.7 dB 25.7 dB < SNR

3.4 Sub-optimal Power Allocation

Since we determined the thresholds for deciding among modulation levels for each

channel and managed to create a practical bit encoding algorithm at the relay, we

can implement the bit loading. On the other hand, we can also distribute power

with careful calculation to users to obtain the most power efficiency in pursuing the

maximum throughput.

As discussed earlier, the water-filling algorithm is the known optimal method to

allocate power, but the bit distribution obtained can be any real number, which is

difficult to implement in practice; while the Hughes-Hartogs algorithm is based on

discrete information bits or a finite granularity, and is used in this thesis for it being

a classic optimum bit loading algorithm in literature [33] [41].

For a multi-channel transmission system, the Hughes-Hartogs algorithm, based

on the greedy optimization approach, iteratively assigns one bit to a subcarrier that

requires the least transmit power with the constraints of total power and a BER

target. The meaning of “greedy” here is that the power is allocated with the goal

to send more bits in the system. Hence, subcarriers only obtain the exact amount of

power to achieve the BER required for their modulation level transmission, and any

extra power is collected for possible increase of modulation levels for other subcarriers.

52

The original Hughes-Hartogs algorithm assigns one bit at one time, however in our

system, the selected modulation types are BPSK, 4-QAM, 16-QAM and 64-QAM,

where the bit increment is one from BPSK to 4-QAM and is two between other

neighboring modulations. Therefore, the Hughes-Hartogs algorithm is revised to suit

our scheme.

3.4.1 Sub-optimal Power Allocation

In this section, we demonstrate the process for performing the sub-optimized

power allocation for achieving better throughput. The power distribution is feasible

through the control of precoding vectors as discussed in Section 2.2. To reallocate

transmit power among users with the target BER, we need to add different constraints

on precoding vectors with the total power limit. Refer to the transmit power con-

straint in (1.2), and when we add parameters to adjust the power among precoding

vectors, it is transformed into:

x · (|v21|2 + |v12|2) + y · (|v31|2 + |v13|2) + z · (|v32|2 + |v23|2) =x

3+

y

3+

z

3= 1 (3.5)

with

x+ y + z = 3 (3.6)

where x, y and z are the power allocation parameters to achieve the sub-optimum

discrete bit loading for the proposed scheme and (3.6) will guarantee the total trans-

mit power remains one. Then, the adjusted SNRinstant,m for each user pair is x ·SNRinstant,12, y · SNRinstant,13 and z · SNRinstant,23, respectively.

Now we have the parameters representing the allocated power for each user pair

with the total power limit, and we will proceed with development of the sub-optimum

bit loading with power allocation are as follows. First, the thresholds for using each

modulation with a target BER need to be defined. The target BER is still selected to

be 10−3, and then the thresholds used here are identical with the ones in Table 3.2.

Next we calculate the power that is required for each user pair with their original

SNRinstant,m to achieve the BER performance using varying modulation levels as:

pm,n = Tn/SNRinstant,m (3.7)

53

where n = {1, 2, 3, 4} represents BPSK, 4-QAM, 16-QAM and 64-QAM, respectively,

and pm,n is the relative power needed compared to SNRinstant,m with nth modulation

and Tn is the SNR threshold to achieve the BER target with nth modulation scheme.

Table 3.3: The relative power required for achieving the aimed BER 10−3 using eachmodulation scheme compared to the current transmit power for each user pair.

- SNRinstant,12 SNRinstant,13 SNRinstant,23

BPSK p12,1 p13,1 p23,14-QAM p12,2 p13,2 p23,216-QAM p12,3 p13,3 p23,364-QAM p12,4 p13,4 p23,4

Table 3.3 lists the ratio of required transmit power to achieve the BER aim with

a certain modulation level and the current transmit power. When pm,n is greater

than one, it states that the channel with SNRinstant,m needs more transmit power to

reach the BER aim with nth modulation; when pm,n is smaller than one, it shows

that the SNRinstant,m is greater than the power that this modulation is required

to achieve the BER target, and hence extra transmit power can be collected to be

redistributed to a subcarrier that is closest to its next modulation level. For example,

when SNRinstant,13 = 8, p13,1 = T1/SNRinstant,13 = 6.9/8 = 0.8625, which means that

only 86.25% power of the precoding vectors v13 and v31 are required to use BPSK

to achieve BER 10−3; p13,3 = T3/SNRinstant,13 = 16.7/8 = 2.0875 means that 2.0875

times the present power of precoding vectors v13 and v31 are needed to apply 16-QAM

modulation with target BER 10−3.

The power parameters in Table 3.3 are obtained by comparing with the sub-

carriers’ original SNRinstant,m, which states the power required for each individual

modulation, but when the subcarrier intends to jump to a higher modulation level,

the power needed is the subtraction of the power parameter of the current modulation

level from the power parameter of the new modulation level, which can be indicated

as the relative required power between modulation levels. Consequently, Table 3.3 is

transformed into Table 3.4.

Finally, the minimum parameter from the set {xa, yb, zc}, (a = b = c = 1

at initial; a, b, c ∈ {1, 2, 3, 4}) is selected and the index of the selected parameter

54

Table 3.4: The relative power required for achieving the aimed BER 10−3 using nextmodulation level compared to the current modulation scheme for each user pair.

- SNRinstant,1 SNRinstant,2 SNRinstant,3

BPSK x1 = p1,1 y1 = p2,1 z1 = p3,14-QAM x2 = p1,2 − p1,1 y2 = p2,2 − p2,1 z2 = p3,2 − p3,116-QAM x3 = p1,3 − p1,2 y3 = p2,3 − p2,2 z3 = p3,3 − p3,264-QAM x4 = p1,4 − p1,3 y4 = p2,4 − p2,3 z4 = p3,4 − p3,3

increases by one each time to represent the power parameter for the next modulation

level. The process repeats until the sum of all selected parameters reaches the total

transmit power limit. The final subscripts of x, y and z indicate the modulation the

corresponding channel should use. If there is a little power left and this amount of

power cannot support any channel to upgrade modulations, it will be evenly divided

to all carriers to boost their performance with the modulation they can apply.

3.5 Simulation Results

In this section, we demonstrate the simulation results for bit loading and the sub-

optimum power allocation with the target BER 10−3 in the MIMO Y channel with

orthogonal signaling. We implement the iterative search for optimized parameters

to achieve better throughput while maintaining acceptable BER and balancing the

three users’ performances. Fig. 3.6 (a) shows the BER for the scheme with bit loading,

while Fig. 3.6 (b) presents the BER for the scheme with the integrated bit loading

and power allocation. Moreover, Fig. 3.7 illustrates the throughput results associated

with Fig. 3.6. Again throughput in this thesis is measured in successfully transmitted

bits per time slot for one user pair.

In the lowest SNR regime, the BER in Fig. 3.6 (a) for all users are not meeting

the targeted level so there are no results/lines in this region. The target BER is

always met when SNR is above 10dB. When SNR is between 7.5dB and 15dB, the

BER shows differences for three users, which seems to be against the fact that our

iterative search is attempting to balance the three users performances. The reason is

that the low SNR in this case causes low throughput, so that the simulation results

may not be as accurate as the ones in higher SNR. Fig. 3.7 demonstrates that in the

low SNR regime between 0dB and 10dB, the throughput of bit loading is almost zero,

55

SNR (dB)0 10 20 30 40

Bit E

rror R

ate

10-5

10-4

10-3

10-2

10-1pair12pair13pair23

(a)

SNR (dB)0 10 20 30 40

Bit E

rror R

ate

10-5

10-4

10-3

10-2

10-1pair12pair13pair23

(b)

Figure 3.6: The BER comparison between the system with (a) the bit loading andthe integrated bit loading and (b) power allocation.

while the throughput of the bit loading with power allocation increases from zero to

around 0.37.

With SNR above 15dB, the bit loading with power allocation (Fig. 3.6 (b)) has

worse BER performance than the bit loading (Fig. 3.6 (a)) . As we indicated in

Section 3.3, if the obtained BER is better than the target, the extra power can be

56

SNR (dB)0 10 20 30 40

thro

ughp

ut

0

1

2

3

4

5

6bit loadingbit loading with power allocation

Figure 3.7: The throughput comparison between the system with the bit loading andthe integrated bit loading and power allocation.

collected and distributed to parallel (spatial) streams which can offer higher through-

put. This is corroborated in Fig. 3.7 where the throughput of the bit loading with

power allocation is indeed better than that for the bit loading alone. This reinforces

the statement that the bit loading with power allocation assigns power and bits more

efficiently by achieving better throughput while meeting the BER constraint.

However, when SNR is around 40dB, BER in Fig. 3.6 (b) and in Fig. 3.6 (a) are

close to each other. Due to such high SNR, the selected modulation for either case

is usually the available highest one, 64-QAM. This implies that, in the bit loading

with power allocation, power cannot be used to further upgrade modulation levels for

any subcarrier anymore but can be used to boost the BER performance with current

modulation levels for each subcarrier. Fig. 3.7 also shows that the two methods nearly

reach the highest available throughput around 40dB of log2(64) = 6 bits. However,

we can see that the bit loading with power allocation offers higher throughput than

the bit loading in the SNR regime in Fig. 3.7. Therefore, the proposed bit loading

with power allocation in the MIMO Y channel is an efficient way to allocate power

to improve the throughput and boost the BER performance compared to bit loading

alone.

57

It is worth mentioning that the Hughes-Hartogs algorithm can have better through-

put performance than its modified version in this thesis, as the power allocation in

the Hughes-Hartogs algorithm is considered bit by bit, while in the proposed scheme,

the bit increment is different in the selected modulations BPSK and M-QAM, which

sometimes may cause inefficient power allocation for improving throughput. For in-

stance, there is a subcarrier requiring power a to transform from BPSK to 4-QAM

and another subcarrier requiring power b to transform from 4-QAM to 16-QAM. If

b > a, the proposed algorithm would choose the first subcarrier, but if at the same

time a > b2, this selection may not be the most efficient as the second subcarrier

needs less transmit power per bit. However, the selection can be seen as the one

that the system tries to encourage all user pairs to transmit even though they are

using BPSK. Therefore, although the Hughes-Hartogs algorithm is better than the

proposed scheme in maximizing the system throughput with the power allocation

considered bit by bit, the proposed algorithm is much simpler with less iterations

and offers sub-optimal power and bit allocation. Moreover, the system only needs

to implement the common BPSK and M-QAM encoders and decoders rather than

complex ones for the original Hughes-Hartogs algorithm.

Chapter 4

Conclusion

In this chapter, an overview of the thesis contributions and suggestions for future

work are presented. Section 4.1 discusses the contributions of this thesis and Section

4.2 suggests the future work.

4.1 Thesis Contributions

There are two main contributions in this thesis that pertain to improving the

reliability and throughput in MIMO Y channels.

The first contribution is the introduction of orthogonality to the selection of

aligned signal dimensions, which simplifies decoding processes at the relay. It also im-

proves the BER performance for the three user pairs with BPSK signalling. However,

the initial approach suffers from the unbalanced BER performance among different

users. To alleviate the problem and improve BER performance further, different

modifications in the original orthogonal signal alignment scheme were developed.

The second contribution is in incorporating higher level QAM modulation schemes

into MIMO Y channel signaling with orthogonality. Specifically, the bit loading with

power allocation in MIMO Y channels was developed to distribute power and to

assign modulation levels on parallel spatial streams with the goal to increase the

system throughput.

In the area of the first contribution, by working with redundant antennas at the

user terminals, this thesis proposes to constrain the aligned signal dimensions to be

orthogonal. Computationally efficient method was developed to calculate the precod-

ing vectors. The orthogonality simplifies the decoding process at the relay offering

the performance of the minimum distance ML decoding. By projecting the received

signals and slicing them along three orthogonal directions, the receiver complexity is

lower than that of the ZF decoder which requires matrix inversion, and the proposed

scheme does not suffer from noise enhancement caused by ZF. Moreover, when higher

58

59

modulations are used at users, the proposed decoding process still has the same com-

putation complexity as when users use BPSK, while applying ML decoding directly

can be very complex due to the size of the superposed constellation at the relay,

e.g., 2253 calculations of distance are required at the relay when users use 64-QAM.

Hence, the simplification of decoding process from the proposed scheme is of great

importance using higher modulations. In addition, as all user nodes and the relay

node have the same antenna configuration in the proposed system, each node can

perform as a user or a relay. Hence, the system can be part of ad hoc networks, or

the multiple systems can constitute an ad hoc network.

In the area of the second contribution, this thesis presented first the process of

bit mapping in QAM constellations used at the terminals in the MAC phase in order

to recover the mutual bits in the expanded constellations at the relay, so that these

mutual bits could be re-encoded in a reduced-size (original) constellation in the BC

phase. Because of the constellation expansion at the relay, e.g. from 16 = (4 × 4)

QAM to 49 = (7 × 7) QAM, there was a need to carefully design the bit assign-

ment to minimize the impact of symbols in error, in a similar way that the Gray

encoding accomplish this objective in the conventional modulation schemes. Once

the higher level QAM processing was established, a method of distributing bits and

power allocation among the spatial streams was developed by working with discrete

bit allocation algorithms similar to those used in multicarrier modulation systems.

Specific problems and detailed solutions of the first contribution are summarized

as follows:

1. Precoding Vector Design for MAC Phase

First, the precoding vectors as in the signal pre-processing at the terminals need

to satisfy the SA requirement to put the mutual symbols from two users in one signal

dimension. If doing so with the minimum configuration of antennas, there is only one

eligible signal dimension for one pair of mutual signals. The obtained aligned dimen-

sions and the effective channel gains for aligned symbols are only SA constrained,

which leads to the situation that the Euclidean distance between neighboring aligned

symbols varies within a large range. The SER performance in the communication sys-

tems is limited by the worst case; therefore, as long as the distance between symbols

60

appears to be small (even occasionally), the whole BER performance will be influ-

enced badly. In order to provide control of the aligned dimensions, in this thesis, each

user terminal is equipped with one redundant antenna, which brings more diversity

gain that can afford to satisfy additional signal dimension requirements such as to

maximize the distances and to improve reliability. We also include the orthogonality

as one of the conditions to select precoding vectors so that the decoding at the relay

can be simplified by projecting received symbols on the three orthogonal dimensions

and thresholds. Using original ML decoding (which calculates the Euclidean distances

between the received signal and ideal symbols) may become complex for higher modu-

lation levels. Our design was actually motivated by simpler implementation of the ML

receiver by sacrificing some performance loss (in terms of BER), as a result of using

hyper-cube constellation of points and higher number of antennas. Computationally

efficient method was developed to calculate the precoding vectors with the orthog-

onality constraints, however, the method introduced unbalanced BER performance

among different users which had to be addressed as discussed next.

2. Modifications and Improvements with BPSK Orthogonal Signalling

The BER results from the proposed MIMO Y channel are not balanced for the

three user pairs due to the preference in the sequence of solving the sets of equations,

i.e., constraints. In order to have even performance for all users, two methods are

applied: one is the power allocation where the transmit power of the user pair with

better BER is reallocated to compensate the user pair with worse BER; the other

one is the iterative search by solving the equation sets in six different sequences of

steps. In the iterative search, the outcome which offers the best performance for

the worst user pair/link is selected from the six sets of solutions to mitigate the

worst scenario. Then, the obtained assignment of effective channel power gains for

different user pairs is randomized, which can balance the BER of the three user

pairs. Next, to further improve the BER performance, an efficient approach called

time scheduling is proposed where users stop transmitting if the channel condition

is not acceptable. This method aims to avoid the worst scenarios by users stopping

transmitting messages in such situations. A small amount of throughput would be

lost, but BER is significantly improved. Moreover, when iterations are combined with

time scheduling, the reduction in throughput of the proposed system over the original

61

MIMO Y channel is not that significant while achieving the targeted BER.

Contributions in the second part of this thesis are as follows:

1. Deployment of Higher Level QAM Modulations

We propose a mapping strategy for using higher modulations in SA. Deployment

of BPSK at the terminals results in the need for ternary signal detection at the

relay (in each spatial dimension) and the proper interpretation of received signals

as the XOR-ed version of mutual symbols. Since the QPSK can be considered as

two independent BPSK modulations along the in-phase and quadrature components,

the generalization of BPSK into QPSK is rather straightforward. In our approach,

M = (L×L) QAM at the terminal is a superposition of two L PAM signallings along

the in-phase and quadrature components. When combining two L PAM mutual

symbols (in each spatial dimension), at the relay we had to detect (2L− 1) PAM (in

both in-phase and quadrature dimensions), which is equivalent to ternary symbols

when L = 2, i.e., when we deploy BPSK. It is this observation that led us to work

with the expanded M ′ = (2L−1)× (2L−1) QAM constellations at the relay and the

need to the bit mapping so that we could recover the mutually bits. However, when

L is greater than 4, the mutually bits cannot be mapped to their XOR results due to

ambiguity. Hence, we propose the mapping strategy that can cancel the ambiguity

with low computational complexity.

2. Bit Loading in the Spatial Channels

In this area of our research, we implement the bit loading with power allocation

in the MIMO Y channels. The algorithm offers the system the capability of changing

among modulation schemes on three spatial (parellel) channels and allocating power

to achieve a higher bit rate, with constraints of total power and target BER. The

bit loading with power allocation used in this thesis is adapted to the underlying

system from the Hughes-Hartog algorithm. In the proposed scheme, the number of

assigned bits in each iterative assignment is not the same, while in the Hughes-Hartog

algorithm which offers optimized throughput, the number is the same. The proposed

algorithm offers sub-optimal performance but is simpler with less iterations than the

Hughes-Hartog algorithm, and a system only needs to implement BPSK and M-QAM

encoders and decoders to use the proposed algorithm.

In summary, the contributions of this thesis are obtained by exercising the tradeoff

62

among system complexity, computational complexity, system reliability, bandwidth

efficiency and power efficiency. In the original MIMO Y channel, the bandwidth

efficiency is the most noteworthy advantage, but the BER performance is lacking even

in a high SNR regime. In the proposed schemes, some degradation in throughput can

be traded off for significant improvement in BER. Specifically, in Chapter 2, after

introducing precoding vectors for orthogonal spatial signaling, the iterative search for

balancing user pair BER performance is combined with time scheduling to reduce the

throughput loss, but this introduces higher computational complexity in obtaining

precoding vectors. Moreover, the adaptive modulation introduced in Chapter 3 leads

to higher throughput with a targeted BER by efficiently allocating power and bits,

but again the system implementation is more complex.

4.2 Suggested Future Work

The potential for future work is discussed in this section.

As in any new cooperative communication scheme, there are some generic prob-

lems such like how to handle the effects of imperfect CSI and synchronization, and

there are some issues specific to our schemes. Even more pressing issue is that our

work is at the theoretical level verified with simulations so it is natural to ask about

feasibility of practical applications. To address the latter concern, there are already

some practical verifications of SA schemes using a real-world communication testbeds

that demonstrate that the observed gains of SA in the practical settings are in accor-

dance with the theory and simulation results [42].

A few topics related to the present study are suggested as follows.

1. Effects of Imperfect Channel Information

In MIMO Y channels, the perfect CSI is assumed to be available to all nodes, as

users and the relay need global channel conditions to calculate proper precoding vec-

tors to implement SA. However, in practical applications, CSI is normally gathered

through receiver estimation and transmitter feedback, which would cause the delay

and inaccuracy of CSI at user nodes. This thesis does not consider this situation,

but this problem has been studied by many researchers in other settings by analyz-

ing performance with imperfect CSI and proposing new algorithms to mitigate this

issue [43]–[45].

63

2. Precoding Design for BC Phase

In this thesis, we only discuss the MAC phase for precoding designs in the proposed

scheme, and for the BC phase we assume that the nulling vectors can be obtained with

antenna selection. However, more detailed investigation can be pursued to exploit the

possibility of improvements in the BC phase.

3. Synchronous Transmissions

SA is about aligning two symbols, or equivalently two synchronous signals, in one

signal dimension at the relay where the addition of synchronous signals is through

the over-the-air summation. The problem is that sometimes it is not sufficient to

be synchronized when sending the data because the requirement is for synchronous

reception. In that case, users are required to coordinate with each other through

network protocols and other methods with regard to their locations, which are not

studied in this thesis. However, this issue can be solved with OFDM technology where

the symbol rate is very low in each subcarrier so that the delay over different user

paths to the relay can be neglected. In our work, we were also assuming that all users

have the data to be sent at all the time, which may not be the case for asynchronous

traffic. This may be solved through different buffering strategies.

4. Path Loss Considerations

Deterministic path loss (or path attenuation) is the reduction in power of an

electromagnetic wave as it propagates through space over given distance. In this

thesis, we did not consider deterministic path loss, i.e., the distances between each

user and the relay are assumed to be the same. However, in reality, the distances

are normally not the same, and the attenuation of signal power is related to the link

distance. Therefore, to let the two mutual signals have the same power weight in the

aligned signal dimension for achieving higher decoding accuracy rate, the distances

between users and the relay should be considered for power allocation.

5. Bit Loading with Power Allocation

The proposed bit loading algorithm provides the sub-optimal solutions because it

does not consider the most efficient power and bit allocation as the Hughes-Hartog

algorithm does, where the incremental bit unit is one between neighboring modula-

tions. However, the proposed algorithm can be improved by selecting the subcarrier

with the least power per bit rather than the least power to upgrade a modulation

64

level to achieve the optimal solutions with current conditions. Moreover, we use the

RA strategy in this thesis to have the maximum throughput with the constraints of

total power and BER, but MA can also be considered for cases where the requirement

is to find the minimum power to achieve a data rate and a target BER.

Appendix A

Average Symbol Energy in the Superposed Constellations

In this appendix, we elaborate on the average superposed symbol energy at the

relay for different modulations used at the user sides. The first observation that we

will make is that if there is no attenuation between the users and the relay the min-

imum distance between the symbols in the original constellation is the same as the

minimum distance between the superposed symbols at the relay. The second obser-

vation is that the average energy of the superposed symbols is twice the energy of

the original symbol constellations. The latter result is quite obvious as superposed

symbols are the sum of two symbols from the original constellations where all origi-

nal symbols are equiprobable. We corroborate this observation by performing more

detailed analysis of the superposed symbols. Specifically, this is accomplished by an-

alyzing the statistical distribution for the frequency of occurrence of different symbols

in the superposed constellation. Based on these two observations, we conclude that

the symbol error rate performance of the superposed constellations is 3dB worse than

that of the original constellation at the user sides.

First, we show that the minimum distance between symbols is the same in the

modulation constellation at users and in the modulation constellation at the relay,

although the received symbols at the relay are two superposed symbols from users.

Assuming the average symbol energy of transmitted symbols at users is equal to one

despite modulation levels, the relationship between the modulation (BPSK, 4-QAM,

16-QAM and 64-QAM) at users and its corresponding received modulation at the

relay is illustrated in the following figures. Each figure shows that the minimum

distance between symbols in a modulation constellation for transmission is the same

as the one in the corresponding modulation constellation at the relay.

65

66

Figure A.1: The minimum distance between symbols in BPSK modulation used atusers and in the corresponding ternary symbols at the relay.

Figure A.2: The minimum distance between symbols in 4-QAM original constellationat users and in the corresponding superposed constellation at the relay.

Figure A.3: The minimum distance between symbols in 16-QAM original constellationat users and in the superposed constellation at the relay.

67

Figure A.4: The minimum distance between symbols in 64-QAM original constellationat users and in the superposed constellation at the relay.

Moreover, the average received symbol energy should be two because the received

symbol is two superposed symbols whose average symbol energy is one. This can be

proved by considering the frequency of occurrence of different superposed symbols and

weighting correspondingly their energies. We take the original 4-QAM constellation

as an example, and the probability of each received 9-QAM symbol at the relay in

the superposed constellation is shown in Fig. A.5. Then the average received symbol

at the relay can be calculated as:

Es,avg = 4× 1

16×

{(2√2

)2

+

(2√2

)2}

+ 4× 1

8×

{(2√2

)2

+ 0

}+ 4× 1

4× 0

= 1 + 1 + 0 = 2

(A.1)

Through similar calculations for other modulation levels, we verified the same

result that the average superposed symbol energy is twice the energy of the original

constellations. In this way, the received symbols at the relay require twice of the

energy of the original constellation to maintain the same minimum distance between

symbols in the superposed constellation. This indicates that the BER performance of

the received symbols is 3dB worse than that of the transmitted symbols at the user

sides.

68

Figure A.5: The probability of superposed symbols at the relay when the equiprobable4-QAM is deployed at the users.

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